See it all the time, especially with teachers--husbands and wives--lovers.

How is it properly handled socially when you both still dance at the same scenes (assuming that things didn't get too bitter)?

I have noticed that usually one couple drops out of dancing, moves, or relocate your dancing if your city is big enough.

In a regular relationship, unless you have kids, you don't see the person that much and you just move on.

But with dancing there they are on they floor with a new partner or friend and you are with a new partner or friend--and it just feels uncomfortable.
Lots of dancers use dancing as their main social outlet or lifestyle and it is a big inconvenience to change studios, teachers, friends and etc.

The loss is not so much the person you were dancing with, but the dancing that you created together.