This one is different. I not only think they have sex during NGE, I think we witnessed part of it, and heard another part unknowingly.
It would be nearly impossible to convincingly summarize this theory in text or image form, which is why I made a video drawing on the necessary footage.
Updating this thread with three new points of interest
1) Based on the contents of "Everything You've Ever Dreamed" and episode 15 in this image (see this thread for more on the subject) it seems that Shinji and Asuka had more interactions on the night of episode 15 beyond what we saw. I've always thought this from the first watching because of that very abrupt "THE NEXT DAY" cut that happens right after she yells at him that she's feeling sour because she kissed him.
What was it she did to break your heart
Betray your heart and everything
Kiss you with a kiss that wasn't true
It wasn't you at all
This is clearly in reference to Asuka using Shinji for the episode 15 kiss while thinking of Kaji.
Shinji had to have figured it out somehow for it to have hurt him, and I don't think he only figured it out when he looked into her mind in instrumentality. Because the song and that image both make it sound like he reached the realization on that same night.
How did he come to realize it? The only way is if Asuka somehow revealed it. A moment of intimacy seems like the correct opportunity where Asuka would let something like that slip. It also can also explain why the encounter might have gone sour, something that originally seemed at odds with Asuka's cool tone in her "Do you love me?" line.
2) There is a past precedent for Asuka saying something to Shinji in English off-screen.
See this thread.
To make it short, the person who said "I need you." (the English line from EoE's final scene title screen) was Asuka, in response to Shinji's comment: "No one needs me, so it would be better if I didn't exist." This is a certain thing, based on the previous placement of that title screen before it was moved.
"Do you love me?" seems to be something Asuka asked Shinji during a moment of intimacy.
Curiously, the line appears in both Japanese (in the Asuka memory flashback) and in English (in the apparent Shinji memory flashback.) https://youtu.be/PcXGgjyV4WA?t=245
So why are off-screen lines to Shinji showing up in English?
Maybe it's because English is being used as "mind to mind language", or whatever you want to call it. So "I need you" doesn't get expressed in plain Japanese when you're communicating the Instrumentality way, mind to mind.
This scene popped into my mind while thinking about this.
More thought is required on this part (why the lines are in English.) But overall this is an interesting connection.
3) Why does the sex seem to take place on a futon considering they have beds and are rarely seen using the futons (once during the Episode 09 sync training, once during Episode 18)? In small part likely because it would be easier to evade drunk-KO'd Misato by using a different room from their bedrooms, but primarily because Asuka was recreating the conditions that she thought would lead to sex during Episode 09 except making sure that sex happens this time.
As this image shows, Asuka was crushed that Shinji didn't breach the "Wall of Jericho."
Here is that scene. She actually implies sex right before the Wall of Jericho line.
Asuka: So where's Misato?
Shinji: At work. She called earlier and said she'll be there all night.
Asuka: So that means we're all alone tonight.
(Asuka enters the futon room)
Asuka: This is the impenetrable Wall of Jericho. Take one step over this wall and you're a dead man.
The way this reads to me is the same way as the circumstances of the kiss. Asuka was awake when she went to lay down with Shinji and he nearly kissed her, her knowledge of what happened the next day confirms that. It seems likely that she stewed on it. Fast forward to episode 15, Asuka straightly asks for a kiss, no subtlety. She forcefully recreates the opportunity that she lost that night by talking about her mother. Amazingly, she actually brings up the topic of their mothers in the second kiss attempt too, but more softly. It seems that it's a topic she wants to talk with him about. Her original tearful utterance of ''mama" was probably meant to bait a question from Shinji on the subject.
Anyway, that's the point I'm trying to make: Asuka stubbornly 'recreates' opportunities she lost with Shinji because she can't stand failure, and I think the sex was another one of those recreated moments. She took him to the "Wall of Jericho" room and did it on the futon, rather than their beds, because that's the original place where she broached the idea and he missed it.