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Clearly they are getting something out of the bickering, or they wouldn't do it.

I guess, but if you're thinking it's attention, I don't think so. All they ever get is "Stop it! Both of you stop! I don't want to hear it!" or varieties thereof. Not very productive/positive/creative, but I don't take sides.

I actually think they are just mad at each other and/or have negative feelings towards each other and this is how it's coming out.

Separation is impractical because it's so constant, and we need to do things like eat together and do activities together and ride in the car. We have gone so far as to impose silence in the car when it gets bad, but it feels kind of awful to do that.