AsuRei

Canon
Originally Asuka and Rei had a very bitter rivalry. Asuka attempts to befriend Rei however Rei declines her invitation to be friends because she “will only do what she’s ordered too”. This is what sparks their rivalry and what makes Asuka start to become jealous of Rei.

Asuka calls Rei either ‘A Doll’ or ‘Wonder Girl’ to which Rei always attempts to correct her. Asuka starts to dislike Rei because she always sucks up to her commanders and due to that everyone respects her meanwhile Asuka tried as hard as she could to even get accepted as an EVA pilot.

However in a way the girls each respect each other but would never admit it. Rei finds Asuka’s independence admirable. And Asuka would do anything to get as much respect as Rei gets. Therefore leading their relationship to be a sorta ‘Respectful enemies’ scenario.

However Asuka’s jealousy seems to die down after she undergoes her character arc. She goes from arrogant to somewhat understanding of others emotions and realises that it’s not all Rei’s fault. However it is unknown if Rei ever changed her opinion on Asuka.

Fanon
The ship is the most popular femslash ship in the fandom for multiple reasons. One reason is because there aren’t many female characters to ship with either of them Aside from Hikari and Mari but since Mari’s age has never been confirmed some fans prefer to ship Asuka with someone who they know is the same age as her. Another reason could be because of the ‘Enemies To Lovers’ or ‘Rivals To Lovers’ tropes that’s popular in anime along with the ‘Opposites Attract Trope’

It is currently the 3rd most written Neon Genesis Ship after KawoShin and AsuShin. However some fans may ship this as a way to pair them off because they ship KawoShin.

Another reason it's popular is because they are the two main heroines and love interests for Shinji aside from Kaworu.

Trivia
They are shown to be intimate in a lot of official arts and even in a figurine