I remember watching Birdy The Mighty(1996) on VHS from my local Blockbuster a long long time ago. It was great. Basically, an alien police woman named Birdy pursues an alien on Earth. While trying to finish the criminal off, the student Tsutomu Senkawa gets killed in the crossfire. To punish Birdy Cephon Altera, she is forced to share her body with Tsutomu until his body is repaired. They can switch back and forth between the two bodies, although Birdy also enjoy joyriding the sleeping body of Tsutomu Senkawa wherever she needs to be… often making him wake up in weird places like traffic. Anyways, there are alien monsters, entrance exams, and Tsutomu’s love interest all packed into this four episode OVA of epic importance! The only sad thing is that’s 4 episodes on 2 DVD’s even Ouch? Ah, the 90’s.
They have since made a new version of Birdy based off of a more recent iteration of the manga, but I’m still pretty partial to the original.