A private disagreement becomes harder to repair when an audience starts rewarding each side.

Two people can usually find their way back from a dispute on their own. Once others are watching, positions acquire supporters, and stepping back starts to cost something in public. The argument quietly stops being about the original matter and becomes about who was right in front of everyone, which is much harder to settle.

Keep a hard conversation between the people actually in it for as long as you can manage. If you need advice, ask one person who will not take a side or repeat it.