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I believe that the kind of person Rei would fit the best is an abstract dominant extrovert. She needs a "giver" who will understand her but instead of also being passive he'd be active and initiative coming up with ideas and stuff that Rei would like, taking her places that she'd like, taking her to spend time together the way she'd like.
Sharing books and drinking tea together is ofcourse not bad, but need more active stuff, I need to give effort and initiative that she will be happy to recieve, and you'll need to put effort to pierce her ice and melancholy.
Like (assumin you're already on good trusty terms)
>Ayanami, I know a nice place I want to take you, don't refuse, it's beautyful and I'm sure you're gonna love it.
Gotta be persistent but gentle, genuinely caring and selfless.
Unlike Asuka who went all like
>Hey you, yes you, you better be my friend cuz it' will be convinient for me.
You gotta be like
>Hey Ayanami, I know a local cafe has great coffe and ice cream, let me treat you some, I'm sure you gonna love it. ... Why? Because you're nice and wan't to spend time with you.
Assuming Rei is antisocial and always super straightforward, I guess she'll value straightforwardness a way more than beating around the bush.