An audience can turn a small disagreement into a contest neither person wanted alone.

Once other people are watching, a disagreement grows a second stake that has nothing to do with the subject. Now there is the matter of not being seen to back down, and that stake is usually larger than the original one. Two people who would have settled it in a minute alone find themselves committed to positions in front of a room. Neither chose the contest; the audience supplied it.

When you feel that shift, move the conversation somewhere smaller. Say you will pick it up later, then actually pick it up. Give the other person a route back that nobody else has to witness.