A discussion while drinking beer with friends:
– How is it pronounced? Is it like ann-i-meh? Or is it like anee-meh? Or what?
– I think it’s pronounced as ann-eem.
– I always forget.
– Me too.
Well, I had to look it up. Anime is pronounced as [ɑnimɛ].
Listen here for an example. Now’s that mystery solved.