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Burst Angel contains examples of the following tropes:
Accent Adaptation - In the English dub, Osaka is pretty much Texas. In the DVD commentary, this was explained as almost being out of necessity because of the strong Western influences of the story. And frankly the Osaka arc seems like it was actively written with Texas in mind.
Also of note is the fact that up until this point FUNimation had tried as hard as possible to avoid doing this in any other shows.
A-Team Firing - In the first episode. Jo and her current mark (a very twitchy purple man) engage in a full-on Dakka contest in an alley. Kyohei ends up cowering in the middle of this leadstorm until his scooter is shot to pieces and the purple guy runs out of ammo.
Author Tract - Starting at six minutes and thirty-one seconds into the first episode, the author gives an anti-gun message. It doesn't stop.
Badass Longcoat - Jo is a Badass Longjacket; Takane's a Badass Long-seifuku.
Bait-and-Switch Lesbians - The manga that serves as a prequel to the series has Meg basically out-and-out in love with Jo and desperately wanting to consummate their relationship at any opportunity. This is very toned down to the point of disappearing by the time the series begins and Kyohei is added. And the same episode where we find out that Sei is a Mafia Princess actually has Meg very interested in the handsome men that have shown up, which leaves her somewhere between this trope and Ambiguously Bi.
Crossdresser - Kyohei has to in order to sneak food (and equipment) to Meg during a few episodes when the latter is infiltrating an all-girls' school.
Crouching Moron, Hidden Badass - Subverted during an episode where Meg has her "hunger" switched around with her will to fight. She spends the rest of the episode mopping up gangsters with impossibly cool kung-fu moves... And then later claims to have no memory of what happened at all.
Fork Fencing: Played with in the Osaka arc. Jo mouths off to Takane one too many times in the Hanshin P.D.'s holding cell, and Takane tries to stab Jo with her bokutó. Eating a meal at the time, Jo catches the sword with her fork and holds it in place with one hand.
Hand Cannon: Somehow, Jo has the necessary arm and shoulder strength to dual wield Desert Eagles without any kind of exhaustion. For reference, these guns are supposed to be held with both hands, and can cause hefty men to fall over if not properly braced before firing.
Heroes Want Redheads - Jo and (red-headed) Meg have elements of this, in the anime, and undeniably in the manga.
Law Enforcement, Inc. - RAPT. Also Hanshin Police Department, Inc., owned by the same company as the second oldest baseball team in Japan, the Hanshin Tigers (thus explaining why tigers are emblazoned on their everything).
Mafia Princess - Sei, though we learn this later. She's also much more benevolent than most other examples.
Magic Skirt - Meg's doesn't do the obvious even when she's hanging upside down from a moving train.
Mind Rape - The brain creature that disguised itself as Nadeshiko (in "Wash This Courtyard With Blood") was doing this to Jo when Meg ventilated it with Jo's BFPistol.
More Dakka - Jo lives by this, as does Meg to a lesser extent.
Jo considers this to be a good approach to get a cat out of a tree.
Non-Action Guy - Kyohei, while talented in the kitchen, is a serious coward.
Off Screen Teleportation: Meg somehow ends up on top of the departing R.A.P.T. train in episode three, despite being shown on the station platform just a few seconds ago. It's rather jarring as it's not played for laughs; it's as if the production team just left a scene out.
Perpetual Poverty: While Jo and Meg were running Jack of All Trades in the manga, they were never able to get enough money to cover their expenses. This was mainly due to Jo's aforementioned More Dakka habit.
Phlebotinum Rebel - Jo. This was a accidental rebellion at first; Jo was blown off the ship where the final Genocide Angel tests were taking place. By the time she woke up in the river in New York, she didn't remember how she got there or what she was. Thanks to the time she spent away from her creators, she then fully embraced the rebel part of the trope.
Power Tattoo - Appears when Jo is in combat. Purple with lighter "shimmering".
Psycho Lesbian - Maria becomes strangely attracted to Meg, and starts to think that she'll be spoils of war if she defeats Jo.
Psychopathic Manchild: What Jo amounts to; a cold-hearted assassin who is only really enjoying herself when fighting and who spends all her free time watching ulta-violent movies in an effort to find some way to amuse herself.
Red Baron: Jo has a bit of a reputation in the Tokyo underworld. Her moniker is "The Angel from Hell".
Refusal of the Call - After the double-kidnapping fiasco in the first two episodes, Kyohei tries to quit the job he accepted as the cook for Sei's group; unfortunately for him, they really liked the meal he cooked as an apology...
Rei Ayanami Expy: A little game for you. Take a look at these◊ two◊ images and guess which one is Jo. We'll wait. One episode has another expy who is even closer to the Rei archetype than the blunt and forceful Jo.
Ridiculously Average Guy - Kyohei is really only there because apparently someone felt that normal everyday Japanese young men wouldn't watch the series if one of them wasn't in it. He's billed as the central character and protagonist, but could be written out completely without even trying.
Rollerblade Good - Django has motorized wheels in its feet for highway travel. When not in use, they look like spurs.
Vapor Wear: Most of the women do not bother with this old-fashioned "brassiere" technology in Tokyo apparently. Meg has a semi-justified reason revealed in the manga series: her skin is too sensitive to wear too many clothes at a time.
The Voice - Bai-Lan (offscreen, relayed through Sei).
Wretched Hive - The "anarchy district" which is the ruins of Shibuya after a massive earthquake wrecked the area. The government abandoned the area, and the police refuse to go there unless something huge happens. It's only populated by cyborg criminals and black market dealings.