The story starts at the final scene of the movie The End of Evangelion. Shinji and Asuka had a small conversation at the beach and then decided to try to survive until more people came back.
They managed to find shelter and lived alone with each other for several weeks, but had a troubled relationship, because they both know everything (the good, the bad and the ugly) about the other.
In one of those nights, they had a discussion and Asuka leaves the shelter. Since those two have the worst luck in the world, after that night they both got captured by the JSSDF that had returned from Instrumentality and were taken as prisoners.
Fortunately, not all JSSDF returnees hated them or wanted to kill them on spot, especially after the events of Instrumentality. Yadoraki and others ensured that they were treated well, even if being kept apart from each other for months.
When Maya came back, Shinji got the courage to write a letter to Asuka and asked Maya to deliver it for him. Asuka got the letter and asked Maya to move her to a cell that was adjacent to Shinji's cell. They had some nice conversations, Shinji got a cello and they got to spend some months in that situation where they were able to talk, but not to see each other. So close, but too far.
Meanwhile, Gendo Ikari, who was one of the fist persons to return to live, gathered the remaining pieces of Eva 02 and managed to send them to Euronerv quarters, in Germany. He also convinced the new governments that restoring the Evangelion and imbuing it with an artificial S2 engine would be necessary to mankind's survival. So, people from Euronerv worked on rebuild Eva 02, which they were able to do in a matter of months, since they were the one's who had build it in the first place.
Since the Eva needed a pilot to be activated because there was no way to rebuild the Dummy System, Euronerv staff asked for Japan to send Asuka back, which triggered an important diplomatic issue. Yadoraki was reluctant, but decided to give that decision to Asuka.
She thought about it for a while and decided to return and pilot Eva, because, by doing that, she would've been able to change their current situation and spare Shinji of the burden to pilot Evangelion again. She left, without saying goodbye to him.
In Euronerv, some older people like Amelie and Avil had already worked with Asuka while she was being trained to be a pilot, and vowed to take care of the girl.
However, Gendo tricked them and managed to do a contact experiment to splice Asuka's soul, sending a part of her soul to the Evangelion core, resulting in giving her partial amnesia. But, even by doing so, the experiment wasn't a full success: the Evangelion was functional, but the S2 engine wasn't, because it lacked more pieces of Unit 02 Eva core.
After the incident, Amelie and Avil, managed to understand Gendo's plan – to control the Eva and S2 engine to rule the world and, probably, try to get Eva 01 and Yui back from space - and created a plan: Amelie would make sure that Asuka would survive and Avil went, with several trusted agents, in a suicide mission to tell Yadoraki about the plan and all the info about the research.
After Yadoraki got the message, he searched for the aid of Yuko and Jima to try to create a weapon able to stop the reborn Unit 02 in the battlefield. The only possible solution was to use the old Jet Alone, but try to switch the AI system to a human piloted system, using the remaining pieces of the old Unit 02 core, to make it able to generate an AT-Field.
While Nerv Japan and the JSSDF where trying to run against time to build the new weapon, Asuka's health and overall condition was getting worse, because of the contact experiment, and she was starting to lose her humanity and her mind, just like her mother. Still, she was deployed in the battle field several times, but Amelie made sure that she would only be deployed to activate her AT-Field, which was enough to ensure military supremacy in the battlefield while avoiding to expose her and Unit 02 to danger.
During that time, Shinji wrote 52 letters to Asuka - that never reached her - and was unaware of what was happening. Until the weapon was finished and he was informed about Asuka's destiny and everything that happened. After an enormous outburst, Shinji agreed to fight for Asuka's sake, and went under the contact experiment, splitting his soul to activate the renewed and improved Jet Alone, becoming partially amnesic too.
So, the day of the confront came when Euronerv's forces, leaded by Gendo, managed to overrule Amelie's authority and deployed Asuka and Unit 02 to the battlefield, to fight against Shinji in the renewed Jet Alone.
Both of them didn't remember anything about the other, but had some flashbacks during the battle. At the end of the battle, Unit 02 batteries dried-out and the Jet Alone got the upper hand when Shinji reached over 100% synch rate, but refused to finish the battle. Unit 02 went berserk, with Asuka also breaking the 100% synch rate barrier, and ripped the core tissue that was implanted in the Jet Alone – including Shinji's entry plug, and tried to merge it within its own core, resulting in an uncontrolled activation of the S2 engine.
That created a Dirac Sea, that engulfed the Unit 02 and the Jet Alone. This event lasted only a few seconds in the real world, but it was a hole different experience for Shinji and Asuka.
Since they both had lost their physical forms due to the high synch rate, their amnesic selves, formed by their damaged and incomplete souls, found each other in a train that was created by their imagination.
Their interactions on the train managed to push them deeper into the Evangelion's cores, which contained the remaining pieces of their souls. They both got the remaining of their souls attached back into them, which made they recover their lost memories.
After that, their collective imagination put the two of them on an imaginary beach, also inside of the Evangelion core. There they realized their situation and decided, after a little dramatic conversation, that they would not return to their human forms. Instead, they would try to control the Evangelion, together, in their current state, to avoid causing the expansion of the Dirac Sea and another end of the world event.
Then the complete Unit 02 with a functional S2 engine ripped apart the Dirac Sea and used that energy to escape Earth's gravity, being launched into the space, while Gendo observed it and waited for his punishment, since his final attempt of being with Yui was frustrated once again.
The destiny of Shinji and Asuka - trapped in Unit 02 - or of the humanity, trying to rebuild the world from the events of Third Impact, is unknown.
