The exact opposite of friends with benefit. It's like having a girlfriend but with all the bad parts and none of the good parts. You go through all the boyfriend/girlfriend drama and there's no making out, no hugging, holding hands, NONE of the good stuff. The worst possible relationship you can have with a girl, worse than being in the friend zone.
Jon: Yo, where's Michael at?
Alex: He's back there with his not-girlfriend, they're arguing about something.
Jon: Offff course they are.
A girl you deeply care about and act like you're dating and everyone THINKS your dating but due to personal or outward forces/choices, you currently can't be boyfriend/girlfriend, but you like to act like you are half the time snuggle-wise.
Tara is totally in love with Justin, but she's eighteen and he's sixteen and she knows his mom will get her arrested, so she's his not-girlfriend.