The game stars Aya Brea, a superwoman who enjoys extended youth despite being a 30+ year old cop and has superpowers thanks to her magical mitochondria. Unfortunately, mitochondria does not come up once in this game, so you just kind of have to infer it/played prior games in the series.
So now you have Aya Brea fighting tentacle monsters that can warp through space and time called "The Twisted." They seem to pour out of a gigantic organic tower that spontaneously erupted in NYC called "Babel." Naturally, some government team recruits Aya Brea and forms the "Counter Twisted Investigation" (consisting of Aya, a jiving black dude named Cray, a woman named Gabrielle, her guide named Hyde, and a computer hacker named Blank) to fight them, led by some FBI dickhead that doesn't trust Aya due to her powers. However, instead of fighting them conventionally, the government somehow builds the "Overdive" System, a computer program that sends Aya's consciousness back in time in order to possess random members of the National Guard in order to fight the Twisted that way. Whenever Aya possesses someone, it only appears to be Aya for gameplay purposes; to everyone else, it's just the same military dude. (We will let slide how this makes no sense in combination with the game's damage "gimmick" where if Aya gets damaged enough, you can potentially strip her to some shreds of her torn clothes and her bra and thong). This "send Aya back in time" plan never works.
After a spectacular failure of her first mission, she returns to the present time and learns that her dickhead FBI boss disappeared, somehow. Her guide, Hyde, is still alive and tells Aya to go back in the past again to fight more Twisted. During this mission, it is revealed through a flashback that Gabrielle actually died a few months ago. Aya does so, fails again, and comes back to the present to find out that Cray was killed offscreen three days prior, and Aya needs to go back in time to save Cray. Also, dickhead FBI Boss is alive again and threatens to kill Aya because her powers are dangerous. Hyde talks him down and convinces Aya to go back in time to save Cray. Oh yeah, Gabrielle is alive again. Aya does not sweat the details, nor does the game really explain or hint why she's alive.
Aya goes back in time three days to help out a random SWAT team and to save Cray somehow. Gabrielle is still alive and decides to be Aya's guide. Aya gets as far as destroying the Babel, but it turns out that there are multiple Babels that house a big shitload of Twisted tentacle monsters. In the present, dickhead FBI boss uses sleeping gas to knock out the team because... he wanted to send Aya to her death for... reasons. Also, Gabrielle turns into a Twisted and Aya is forced to kill her. In present day, Gabrielle wakes up, and is about to kill FBI boss, but she suddenly vanishes because Aya killed her three days prior when she turned into a Twisted. Man, what can Aya possibly do when considering that FBI boss is about to get his way and remove his most feared obstacle to [insert goal here], Aya Brea!?
... Aya travels back to the present and Cray (who is alive again despite how Aya did nothing to really save him) informs Aya that FBI Boss suddenly disappeared and Aya can do whatever the fuck she wants. That's all the explanation you get; he's just gone from the story after this point. The datalog implies that he was literally kidnapped by Russians. At this point, Hyde tells Aya to go overdive into a National Guard team that attacked Babel last and Cray has a hissy fit because of the idea of using Aya for warfare (?!). She attacks the Babel and there is a very bizarre subplot of past-Cray killing the National Guard team, impersonating the Captain, and threatening to kill Aya so he can be with his daughter again or some shit. Past-Cray tells Aya to look for her sister Eve, who Aya doesn't really remember. Past-Cray merges with the Babel and dies, and Present Cray disappears.
And then, suddenly a timeskip occurs. Aya is now rooming with a Parasite Eve 1 character, Maeda!
