We have the same problem too. Both are just each provoking the other one pretty much on a loop especially when they are tired. We are taking steps to try and minimise the arguments.
Things that worked for us:
Every time DD1 comes home from school and starts quite deliberately an argument with DD2 I ask her to come out of the room with me, we sit down in private and I ask her why she is feeling so angry. Eventually she will come out with the reason and there almost always a valid and sometimes important reason; something at school, all the usual stresses with friendships etc, we talk about it, I allow her to cry if she needs to, to get the stress out of her system and we hug it out. She always comes back in better and calmer mood, and sometimes she apologises to DD2 for being grumpy. This seems to help avoid particularly after school conflict, as DD1 doesn't always tell me that she has had a bad day but it comes out in this kind of behaviour.
We have now built in time for our dds to have afternoons and days out together each week usually or at the very least every fortnight. When they are having fun together, they stop arguing and remember how much they like to be together. It reminds them that their relationship is just as important as all the others in their lives, and it also seems to keep them close.
We have gone back to time out, sending both to their rooms when they start bickering until they are feeling in a better place, and then they came come back down and join us. I am refuse to live in open war fare.
When one starts the backbiting I ask them to start the conversation again in a different way that isn't going to be goading in nature. We have kept this up religiously, and we get the sighs and eye ball rolling, but then they do revise their words eventually, and the other child is also getting a lesson in how to start a conversation and not an argument.
We ask them if they want a happy gentle family or an aggressive unkind family that argues all the time. They both always choose the first one, because every child wants to live in a happy home even if they are going through emotional moments. We remind them of this choice when they start another argument, I always tell them they are taking us firmly into 'unhappy family' territory and no one wants to be seen to be doing that, so it sometimes works well.
Also I have on some level just accepted some arguments are natural and normal and part of family life. I just go outside and leave them to it if all else fails. One or other usually comes to join me, and we change the subject and the mood by default. If there is no audience then they usually resolve it themselves one way or another without the need for me to referee.
I am not sure any of these tips are particularly helpful, but they have definitely worked for us.