I don't have the script so I can't say for certain, but I'm pretty sure that Shinji ends up with.. *drumroll* neither.
The Rei fact has always squicked me out because, hello, she's a clone of his mom. The fact that Anno left her in as an actual love interest speaks volumes for the extent of his own Oedipus complex...
So by process of elimination, Asuka is the likely choice for me, but I don't think Shinji is MANLY enough to handle her. They have potential, I guess, but Shinji's always so much of a wuss by the time the series ends that nothing actually happens. But what the hell, if Anno made the franchise to fap all over it, pair Shinji up with Rei. Makes sense to me!