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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 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!
spoiler (click to show/hide)
Oh yeah, Aya's team was wiped out off-screen when Aya got done with that Past-Cray/Present-Cray nonsense by Kyle, the Parasite Eve 2 equivalent of Carlos from Resident Evil 3. Kyle now has superpowers for some reason. Aya goes off and hijacks a random National Guard member (Maeda has an Overdive system of his own set up in his apartment living room somehow), fights Kyle (who turns into a super-twisted called a "High One"), and all Babels across NYC merged into one massive structure called the Grand Babel.
In the Grand Babel, it is revealed that the true villain all along was Hyde! He manipulated Aya to fight and kill the Twisted just so that the Babels would eventually merge or something, which would create the Grand Babel (which is actually an Overdive System itself, somehow), just so he could go back to a place called "Time Zero." Hyde is also a "High One" Twisted too. Aya beats Hyde, and follows him back into Time Zero, who Hyde boasts as the "birthplace of the Twisted."
It turns out that Time Zero is actually point where Aya was going to marry Kyle. But that was cut short when a random SWAT Team busts in and kills both Aya and Kyle. But remember Eve? Yeah, Eve (the same Eve from Parasite Eve 2, who was actually a child clone of Aya who was designed to control humans who mutated into grotesque forms due to out of control, artificial mitochondria) was also in attendance. Due to the shock of seeing Aya die, Eve spontaneously got the power to Overdive; she Overdived into Aya, which in turn shattered Aya's soul. The fragments of Aya's shattered soul created the Twisted tentacle monsters which could travel time and space (and killed humans indiscriminately). Also, Hyde, Cray, and Gabrielle just happened to be at this wedding despite how Aya did not really know any of them yet. Because they were there, the shattered fragments of Aya's soul turned them into High One Twisted! So you were not playing as Aya all along, but her underage adopted clone sister in a 30 year old's body. Eve/Aya fights and beats Hyde, and Eve/Aya gets the chance to travel back to the start of Time Zero!
So the SWAT comes in, shoots up Aya, but before the SWAT can kill Eve, Aya gets up and headshots every single SWAT guy in the room. Aya gets up just fine despite being riddled with bullets (and there is no explanation for this). Instead of simply just taking Aya to a hospital, Aya says that she needs to be killed by Eve anyway. To this day, like the SWAT Team appearing out of nowhere with no explanation (some people speculate that Hyde overdived into himself in the past in order to arrange the sequence of events that would turn himself into a High One), people still do not know why Aya even needs to die at this point when considering that Aya is still alive, thus there's no danger of Eve forcibly overdiving into Aya's body. Eve and Aya switch bodies and Eve (in Aya's body) shoots Aya (in Eve's body). Thus, the twisted are never formed and an underaged Eve (in Aya's body) marries Kyle in Aya's place. Huh.
