Mad Bull 34

Published between 1986 and 1990 in the "Young Jump" magazine in Japan, then picked up again in 1999, the stories revolved around fresh young rookie Daizaburo "Eddie" Ban being sent to New York's toughest precinct, the 34th. Here he meets the man who's been assigned as his partner, John "Sleepy" Estes (a/k/a "Mad Bull"). At first the two couldn't be more different if they tried. Eddie is the more by-the-book type of cop, while the only book John probably uses would be promptly bludgeoned into the skull of some random perp, then followed up by half a clip of bullets. Yet despite this, they eventually come to an understanding/mutual respect (much to the chagrin of their superiors) as they tackle a revolving door of increasingly over-the-top villains.

A series of OAVs was released in 1991, and as you can imagine they're pretty violent. The anime unfortunately only represents four of the many stories of the manga. It also leaves out the only lovable villain that is not killed seconds after being introduced: a diabetic electrician named Nickel whose calling card is pissing sugar all over your car.

By the way, this series is by Kazuo Koike.

Provides examples of:

 * Anti Hero: Sleepy, ridiculously so.
 * He seems to be a Type IV.
 * Action Girl: Perrine Valley.
 * Arson Murder and Lifesaving: Seems to only happen once, in the first installment, at the end, when Sleepy's questionable actions are brought into question by the higher-ups, and the only reason they don't strip the big guy of his badge (and presumably set it on fire) is an impassioned speech by Daizaburo.
 * Assassin Outclassin: A running gag in the manga is every single hitman in eastern America has their eyes on taking out the famous "Mad Bull" Estes. These guys are usually black men with various degrees of heavy automatic weaponry; though every now and then Ninja Hookers or Diabetic Tinkers get involved. Nickel is the only one who's survived to try more than once.
 * Attempted Rape: A common occurrence in the world of Mad Bull. The rapists tend to end up on the wrong side of Sleepy's wrath, though.
 * Except, of course, the time that Sleepy decides the best way to let a girl down easy is to try to rape her to convince her to back off.
 * During his review of the anime, Bennett the Sage keeps a running counter of "Blond Women in Peril". It's up to six by the end of the second episode.
 * In the first episode, Sleepy figures out Eddie is a virgin, and hires about a dozen prostitutes. Or he would've, if they hadn't all decided to do it for free once they learned Eddie was a virgin, despite his vehement protests. They literally pile on top of him, and only relent--from disgust--when they learn he's saving himself for marriage. They already had his pants open.
 * Batman Gambit: When Sleepy is forced to relinquish his right to use firearms and Daizaburo is transferred to Perrine as her partner, he eventually makes a bet with Da Chief. If Daizaburo claims that a Corrupt Cop shot someone in self-defense, Sleepy leaves the force. Otherwise, Sleepy gets to join Daizaburo and Perrine as their partner, the right to use firearms again, and to use the precinct's helicopter anytime he wanted. Da Chief accepted, thinking that Daizaburo was in the dirty cops pockets. He wasn't.
 * Big Applesauce: Mixed a bit with "Did Not Do The Research".
 * Big Guy Little Guy: Sleepy's a pretty big guy and the obvious muscle, while Daizaburo's much smaller and a bit smarter.
 * Boom Headshot: People get their brains blown out quite a bit, usually by Sleepy. Sometimes he does something different, like shoot a guy in the neck so many times his head falls off.
 * Brooklyn Rage: Daizaburo's got a thick accent in the dub, and definitely has a hot temper once Sleepy rubs off on him.
 * Buddy Cop Show: Almost a parody of such aspects. Though, as mentioned before, it's hard to tell.
 * By the Book Cop: Daizaburo at first. Eventually, though, he finds some weird respect for Sleepy's sociopathic approach to police work.
 * Clothing Damage: Sleepy, Daizaburo, and a couple of women get most of their clothes blown off by an explosion in the OVA but are otherwise unharmed.
 * Cluster F Bomb: Perhaps in some weird attempt to make it sound more... "American", the English dub is absolutely full of this trope. Again, like many things on this list, this is especially so when done by Sleepy.
 * Corrupt Cop: When Daizaburo is to testify against one who shot an innocent, a group of them threaten to make sure Daizaburo will not get any backup in shootouts (and possibly worse) if he doesn't claim that it was in self-defense.
 * Cowboy Cop: Sleepy takes this trope and breaks it over his knee. It's hard to tell if he's meant to be a parody of such characters or is meant to be taken serious by the writers.
 * Crazy Prepared: Sleepy prepares many traps for criminal to fall into if they catch him unawares.
 * Crouching Moron Hidden Badass: Nickel, in the manga, proves eerily competent at killing people who aren't Sleepy Estes. This is surprising since his first appearance is as a bumbling weakling that pisses on things to let people know he is around, yet the guy has nearly killed Sleepy at least five times.
 * Cut and Paste Translation: Each episode of the OVA is made up of around 3-4 chapters of the manga, with various bits obviously grabbed and used while others are ignored. Originally, Nickel appears more often (and does not use that stupid "choose a can" trap) and proves very competent; the drug boss in episode 2 was a random Bond-styled villain; the scene in which Perrine and Eddie are captured is completely different. The manga actually makes most of the plot make sense.
 * Da Chief: Subverted. The guy Sleepy and Daizaburo work under constantly calls Sleepy out on his actions but is far from being by-the-book, knowing about drugs hidden in his offices by the guys who held his position last and is willing to overlook corrupt cops and assassins as long as they get Sleepy out of his hair.
 * Did Not Do the Research: You get the sinking suspicion that the writers of this OAV had probably never set foot in New York, much less America in general. New York is presented as this big crazy cesspool of crime, rape, violence, drugs, hookers, rape, and very little else. Did I mention rape? The view of the American justice system seems to have been gleaned from 70's and 80's exploitation movies or the dirtiest of noir crime stories.
 * Despite this aspect, the level of detail in cars, backgrounds, guns, etc. is pretty impressive. (See Shown Their Work below.)
 * Dub Name Change: The black assassin "Curtis" in the first episode of the OVA was originally named Black Rose in the manga. In addition, the way he kills his hooker friend is changed in the OVA; originally it was even gorier, since instead of a Molotov mixture, the soda bottle contained various acids.
 * Eagle Land: Some weird bizarre mixture of the two types.
 * Expy: The big opponent of the fourth installment ("Cop Killer"), pits Sleepy and Daizaburo against what can best be described as an armored mixture of the "Predator" and the Xenomorph from the "Alien" series.
 * Sleepy must be what would've happened if Mike Haggar became a pervy Cowboy Cop instead.
 * Genius Bruiser: Sleepy is a lot smarter than his attitude and size would suggest.
 * Gorn: The people populating Mad Bull's world seem to possess very little in the way of bones, as they tend to explode like over-ripe ticks into showers of blood and not much else.
 * Heroic Sociopath: Again, Sleepy exemplifies this trope as well. He's overkill personified. "Excessive force" doesn't even BEGIN to describe Sleepy's conduct at times.
 * Hyperspace Arsenal: Sleepy somehow has access to all manner of weaponry for a simple beat cop, which he uses to brutally destroy even the most minor of crooks. Not to mention the various caches of guns and traps he leaves in his territory. The piping in his room has been converted to shotguns.
 * Improbable Use of a Weapon: Related to above: if Sleepy has no other way of assaulting the bad guys from a high vantage point while wearing civilian clothes, he'll prepare for the occasion by tying around 20 hand grenades to his pubes.
 * Incredibly Conspicuous Drag: The "Mad Bull" himself, an eight-foot-tall muscular behemoth of a police officer with a full beard and mustache, goes undercover as a nun and as a female prostitute. He somehow expects this disguise to work.
 * Jerk With a Heart of Gold: Sleepy. A lot of his crazy actions have some weird underlying "cause" that's meant to justify them, to the point where even Daizaburo, Mr. By-the-Book, realizes he can't help but respect this crazy bastard he's been paired up with.
 * Karma Houdini: A weird protagonist example. Sleepy brutally murders common crooks and rapists, in broad daylight, and usually in front of hundreds of witnesses, and he's STILL got his badge by the end of the day. And that's just the tip of the zany over-the-top iceberg. Oh, and he's the HERO.
 * In the manga, Sleepy spares Nickel the Electrician several times. This is the only assassin that has come close to killing him more than once, and arguably the only person still standing that is an actual threat to Sleepy. Yet every single time Sleepy has a gun to the guy's head, Sleepy spares him because usually Nickel's sugar piss is easy to smell coming.
 * Keep Circulating the Tapes: As of right now there is NO DVD release for any of the Mad Bull movies.
 * The Lancer: Daizaburo.
 * Mad Eye: When Sleepy gets mad...
 * Made of Iron: Sleepy, which helps with all the assassins after him. The guy can take shotgun blasts to his lims and still keep going.
 * Not to mention when the time when Nickles rigged a doorknob to shock Sleepy with a ludicrous amount of voltage, and was still able to follow him to his bosses and kill them all.
 * Moral Dissonance: Sleepy brutally murders hundreds upon thousands of criminals, including petty ones... yet when Demeyer extracts information from a drug pusher by threatening to tell his son what he's been doing, Sleepy looks at him like he's the worst person alive.
 * For good reason. Demeyer was going to drag the guy through his kid's school letting everyone there know that he was a criminal, and this would've happened. Think about what would've happened to the kid
 * New York Subway: The final battle of the first OVA is held there.
 * No Kill Like Overkill: "If they can still stand, then you gotta keep pulling the trigger till they lie down."
 * Odd Couple: Sleepy is a big perverted brutal psychopath with a "heart of gold", while Daizaburo is the smaller, more traditional by-the-book nice guy who tries to keep Sleepy from engaging in his random acts of violence.
 * Police Brutality: In this world, it would be no surprise if you looked up the phrase 'Police Brutality' and found a picture of Sleepy. He's like if the Punisher was a cop.
 * Plot Armor: Sleepy survives an explosion despite being right in the center of the blast.
 * Refuge in Audacity: Sleepy is this trope.
 * After getting stabbed close to his heart, he found time to fuck the nurse in the operating room right after his surgery. This is just the tip of the iceburg.
 * Samus Is a Girl:
 * Shot in The Ass: Happens to Sleepy in the second anime installment while he and Daizaburo attempt to stake out the docks. He's a huge man being shot at by a dozen crooks, so he gets about six bullet holes in his behind. Ow!
 * Shown Their Work: Despite their obvious lack of knowledge/experience with how things in America work, the anime has incredible attention to detail when it comes to cars, backgrounds, guns, etc.
 * Suicide Attack: A cabal of assassins targets a young reporter being protected by Sleepy and Daizaburo; their motto is that they're the best because they don't care if they throw away their own lives to ensure the death of the target. End result: a whole lot of dead assassins. Maybe there really is something to that whole "self-preservation" thing...
 * Tank Goodness: In the end of the third OVA, the last assassin comes in with a tank to blow away our protagonists. Sleepy's response? Lift it.
 * Took a Level In Badass: Daizaburo becomes almost as good as Sleepy at killing criminals.
 * Unflinching Walk: Wearing a cocktail dress!
 * Wrench Wench: Perrine shows herself to know a bit about mechanical design.