Hiding part of mesh

I am making a game that revolves around the player being inside "domes" sort of like a biodome (google "eden project" pictures if you don't know what i mean)

the problem is that the player needs to be able to create new domes through the game. But when a new dome is created it overlaps with the existing one: What i want to do is to hide the parts of the dome that is inside another dome. I made this illustration to explain my problem: If anyone has an idea about how to make something like this it would be most appreciated.