Arpeggio of Blue Steel -Ars Nova-

Arpeggio of Blue Steel -Ars Nova- is a 2013 anime adaptation of Arpeggio of Blue Steel (plus a Compilation Movie, -Ars Nova DC-, and a finale, -Ars Nova Cadenza-, in 2015).

In 2039, humanity is cut off from the sea by the Fleet of Fog, a fleet of incredibly advanced robotic warships (which superficially resemble warships from World War II) that intercept and destroy any vessel that intrudes into the water (or the air or space above it). Without sea commerce, humanity is doomed to a slow, lingering extinction, but in 2056, there is hope with the invention of the vibration warhead torpedo, a weapon that can annihilate a Fog warship in one strike. However, Japan lacks the resources to mass-produce them, so they contract the Blue Steel, a group of mercenaries with a Fog defector (the submarine I-401, known to the crew as "Iona"), to carry the warhead prototype across the ocean to America, where they will be mass-produced to turn the tide.

Tropes used in Arpeggio of Blue Steel -Ars Nova- include:

Absurdly Powerful Student Council: The Fleet of Fog has one, charged with enforcing discipline against members of the fleet who are "infected" by human error and violating the Admiralty Code. They wield corrosion warheads, which is quite a bit more firepower than even the most absurdly powerful human student council.
Aliens Steal Cable: The Fog based their mental models on "official documentation" by humans, which is why they all chose female avatars and why their discipline enforcement fleet dresses like a Student Council.
Arc Words: Where do we come from? What are we? Where are we going?
Art Shift: The Concept Comm System is an austere, 2D-animated world, whereas the real world is (usually, though not 100% consistently) based on 3D animation converted to cel-shading. The CCS also shifts over time, growing from an austere gazebo in a white void to a natural setting as more of the Fog ships using it develop human emotions. It then becomes a Room Full of Crazy after Kongou's Villainous Breakdown. Also, when a superbattleship hosts the CCS, it is a much larger space than when a 'mere' fast battleship or heavy cruiser is hosting.
Color-Coded for Your Convenience: The code rings that surround a Mental Model when they're using their abilities are the same colors as the highlights of their ships. These markings may literally be for the viewer's convenience, representing internal computer states, and not "real" in the show; when Hyuuga and Kirishima pilot two of Iona's active decoys, I-400 and I-402 comment that their quantum signatures have merged, and that they will have to destroy all three to be sure they get Iona, even though they are clearly marked in different colors.
Conspicuous CG: Intentionally used to emphasize the inhuman nature of the Fog ships.
Friendly Fandoms: With World of Warships and Kantai Collection (as they're all about WWII warships, in one way or another), which developed into official collaborations. WoWS added Fleet of Fog skins to several classes (e.g. Atago could be dressed as "ARP Takao"), and KanColle had an event in late 2013 where players could recreate the battles from the series and recruit a few of the characters (Iona, Takao, and Haruna) temporarily.
Gecko Ending: Cadenza is a "finale" for the series, more-or-less, which resolves plot threads like the nature of Shouzou Chihaya, not to mention the war with the Fleet of Fog itself in very different ways from the manga.
Humanity Ensues: The unintentional side-effect of creating Mental Models and interacting with humanity.
Humanity Is Infectious: Every single Fog ship that Gunzou and Iona come into contact with is changed from the encounter. Not always for the better. When Kongou meets Gunzou for parley, she leaves her core behind on her ship body (rather than bringing it in her Mental Model as the others do) in the hopes of avoiding it. It doesn't work.
Humanity Is Insane: Kongou forms this opinion as she sees each of her fleet-mates come into contact with Gunzou and Iona and develop "errors".
Humans Are Special: The Fleet of Fog created Mental Models for themselves in an attempt to grow and develop the adaptability they observed in humans during the course of their blockade. For some of them, it worked a little too well. (Funnily enough, most of the human characters in -Ars Nova- are pretty flat and static -- there's less than 6 hours of total screen time in the anime and sequel movie, and most of the non-battle time is spent on the Mental Models and their interactions and growth.)
One-Gender Race: All Mental Models are female because "official documentation" refers to ships with female pronouns. (Presumably, whichever proto-Mental Model it was that studied the documentation was only looking at English and Japanese documentation -- in German and Russian, some ship classes are grammatically neuter or male, and are referred to by those pronouns.)
Robot Names: Iona comes from the designation I-401 (I-yon...), as Japanese submarines don't get "real" names. Likewise, Fog submarines don't normally get mental models.
Shipper on Deck: Hyuuga ships (heh!) Takao with Gunzou (not so much for Takao's sake, but to get Gunzou away from Iona so that she can have her -- making them Ship Mates).
Shout-Out:
- Kyouhei comments "That's no battleship!" when he sees Kongou's monstrous airship form -- which itself superficially resembles the Death Star.
- Some of the show's visual style appears to inherit from Neon Genesis Evangelion:
  - The flashing text in the opening (much as NGE had "Absolute Terror Field", "Third Impact", and the like, Ars Nova has "Human Extinction", "Mental Model", "Vibration Warhead", etc.)
  - The battle theme heard in e.g. the battle against Haruna and Kirishima sounds like The Beast II.
  - The Conspicuous CG used on the Fog battleships and their Klein field manipulations visually resembles the Angels and their AT Fields, especially as seen in Rebuild of Evangelion.
Single-Target Sexuality: Takao only has eyes for Gunzou, and Hyuuga for Iona.
Something Else Also Rises: When Takao fantasizes about Gunzou becoming her captain, a foghorn sound plays.
Spaceship Girl: Called "Mental Models", the Fog ships formed them to attempt to learn from humanity and improve themselves. They got quite a bit more than they bargained for.
Villainous Breakdown: For all her devotion to the Admiralty Code and her attempts to avoid being infected by humanity, Kongou suffers a pretty severe one in the final third of the series, at first attempting to deny that she has an obsession with Iona (in spite of the evidence to the contrary), and then taking a flying leap off the edge when she is relieved of command by I-400 and I-402, eventually breaking free of her lockdown by sheer willpower -- when she would have scoffed at the notion earlier in the series -- and destroying the American Fleet of Fog so that she can destroy Iona herself and destroy herself in the process, as she has "succumbed" to the same "error" she saw in the rest of her fleet.