So I got the idea in my head of Barbara being the first person Kori met when she arrived on Earth.
Barbara is out patrolling one night and spots what looks like a meteorite crashing outside of Gotham. She heads to the sight and met’s Koriand'r, who pretty much crushes her lips against hers so she can learn the language. After the shock wears off and Barbara starts breathing again, she decides to help Koriand'r out and takes her home.
Which results in her hiding a 6'4" alien princess who eats all her food, brings home strays and can shoot destructive starbolts from her hands, in her apartment, all while trying to ignore the slow building feelings she has for her.