Monogamous non-relationship

A situation in which two people decide that they want to be sexually exclusive without taking on the title of "boyfriend" & "girlfriend" or having any type of official commitment. Often following a period of transition between casual dating and an established romantic relationship.

"John has been with that girl for a few years now. Is he ever going to start introducing her as his girlfriend?"
"He's afraid of commitment and happy being in a monogamous non-relationship. I don't think they are going to put a title on it any time soon."