Ordered Apology

When Jack insists that Tom apologize to Harry.

This trope comes in two forms, based on where he gets the ability to insist:
 * 1) Jack is the Leader, or parent, or other authority figure and insists on it to smooth over relations. (Justly or unjustly. A Beleaguered Bureaucrat or a Corrupt Corporate Executive frequently insist on it because it seems the quickest.)
 * 2) Tom is deeply apologetic to Jack, even though Harry was far more gravely wronged by the matter. Usually Jack refuses to accept his apology until he does it. This is often Jill insisting that Tom apologize to Jack when they are in a Love Triangle.

In both cases, expect a deeply grudging apology that will manage to keep things going for at least a little while -- with Teeth-Clenched Teamwork. Jack may put his foot down on a Backhanded Apology. Then again, he may not.

Anime and Manga

 * In Bleach, Ichigo's little sisters demand that he apologize to their cousin Tsukishima after he punches him. The messed up thing?.

Film

 * Wanda (the woman, not the fish) in A Fish Called Wanda demands that Otto apologize to Archie for assaulting him. This is while they're technically on the same side.

Literature

 * In LM Montgomery's Anne of Green Gables, town busybody Mrs. Lynde comments on Anne's red hair, which leads to Anne flinging all sorts of insults in her face. Marilla is horrified and orders her to apologize. Anne then presents Mrs. Lynde with one of the most melodramatic apologies ever thought up.
 * In LM Montgomery's The Blue Castle's Backstory Valancy was ordered to apologize once to her cousin for something she didn't do. When Valancy is rude to her during the course of the book, her mother tries again, and Valancy says that that apology would have to do.
 * In The Vor Game, while regaining control of the Dendarii Free Mercenaries, Miles Vorkosigan meets a demand that Tung apologize to another officer. Miles insists on it, and then insists on doing it for real after a Backhanded Apology.

Live Action TV

 * The Big Bang Theory: Leonard and Penny have an argument, and Sheldon tries to get each of them to apologize to the other because their fight is disrupting his life. He doesn't really care who apologizes to whom, he just wants the fight to stop.
 * In another episode Penny tells Sheldon to apologize to Leonard for something, but he feels he has nothing to apologize for because he's right. She suggests that he apologize "sarcastically;' since he doesn't do sarcasm well it comes off like a regular apology.
 * When Penny finally chews Howard out for his lechery, he has an emotional breakdown and locks himself in his room. The A Plot of that episode focuses on the rest of the guys needing to amp up their robot, for which they need Howard, the engineer. Leonard forces Penny to apologize to Howard, though he comes around too late to fix the robot.
 * Played with in the pilot episode of The West Wing; after making a rude comment to a religious right figurehead about the lack of intelligence of her and her followers on national television, Josh is basically ordered to apologise to her by everyone he comes into contact with that day, and reluctantly agrees to do so despite feeling he was right and justified. After his apology, however, the woman's arrogance starts rubbing Toby up the wrong way until she makes a rather snide comment he interprets to be anti-semitic, at which point he starts getting into it with her, and the meeting quickly breaks down until the President basically shows up rather awesomely, reveals that his daughter has been targeted by pro-life extremists, and then tells the religious right representatives in no uncertain terms that there'll be no apology from the White House and, until they denounce the extremists within their ranks, they can pretty much get stuffed.
 * On Babylon Five, Sheridan is ordered to apologize to the Centauri for for political reasons. He prepares a truly epic Backhanded Apology for them. It's just too bad he never gets to deliver it.

Webcomics

 * In Blue Yonder, Lena doesn't want Kevin's apology, she wants him to apologize to Jared.

Western Animation

 * In an episode of Allen Gregory, Allen Gregory found his biological mother and thought she was great until she made him apologize to his teacher.