If anyone else runs into this, what I ended up doing was destroying the store on appearance of the "Login" page the user is taken to upon logging out. I didn't do anything special to make sure it only happens in that specific scenario since there is nothing I store in CoreData while the user is logged out, so it is always safe to call that if the user is logged out.
Topic:
App & System Services
SubTopic:
iCloud & Data
Tags: