Rei's constantly dogged by bad luck—turns out she's an Atarunian, an alien species cursed with perpetual misfortune. She landed on Earth hunting for a special spherical stone she'd lost on a previous visit; the thing basically keeps her bad luck in check by balancing it out with good fortune. She tracks down Fuyuki, who'd found the rock back when he was a kid, and gets close to him to get it back. When she leaves Earth, she erases his memory of her entirely.
Momoka absolutely hated Rei over this, convinced she was trying to steal Fuyuki's affection. But honestly, Momoka would've done literally anything to keep that from happening anyway—it was kind of her whole thing.
Content compiled by AnimeList.moe from publicly available sources.