possessives - "John and Me's ping pong game"

If you played someone in a game of ping pong, and wanted to refer to the game later, you could refer to it as "our game". For instance:

"It was after our game."

But if I were talking to a different person about this, I might find myself inclined to refer to the other person and me specifically, such as:

"It was after John and me's game"

But that doesn't sound right and also doesn't quite make sense. I almost more want to say:

"It was after John and my's game"

But I don't think that's correct either. Despite this, I don't feel like expressing it like this should be completely impossible. Is there any way to convey something about a possession that belongs to two people in this way?