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2.
Sharing time and/or space with someone whom you moderately appreciate or have absolutely no respect for to the point of not remembering or deliberately forgetting their name.

Pouet: So, who you out with last night?
Tuatil: Arh! A bunch of jerk-offs. I wasted a whole evening with a load of idiots.
Pouet: Really? Who?
Tuatil: Can't remember. I'm on a no name basis with'em.

or

Rebacci: How's your new roommate Jelad?
Jelad: Alright I guess. We're on a no name basis which shows how much time I spend at home with her!
by Berg Sloe September 10, 2007
 
1.
When you become so comfortable with someone that you no longer even refer to them by their first name. I.e. being beyond first name basis.
Man: "Honey, I'm home!"
Woman: "Hey baby, how was your day?"
Man: "It was great sweetheart."
Woman: "What's my name?"
Man: "Ummmmmmmmm..... sweetheart?"
Woman: "What, are we on no name basis now?"
by Rothdaddy October 30, 2006