The conversation may start off calmly enough, but when emotions get involved, things can flare up. And rather quickly. Not leading anywhere positive.

Hopefully, one of you will notice this before it gets too hot, and will call a ‘time out’. –>“Let’s not let this get out of hand and say things we will both regret. I want to take a step back here.” –>”This conversation is important to me, but right now I’m too emotional to finish it. I’m afraid that one of us will say something we can’t take back. Let’s talk about this in an hour after I gather my thoughts.” Catch yourself mid-sentence, and just call a time-out.
While this may be one place where it is OK to interrupt in a conversation, it may be better to wait until they are finished so as not to add new fuel to the fire.