Do you ever have Significant Others get upset with you for not taking their side in an argument with a 3rd party?
Ex. Mary (your SO) is in an argument with Neil Degrasse Tyson about the earth being flat. Mary is a wonderful person in all other ways but she is a flat earther. You, being the intellectual you are, know this idea is bananas. You tell Mary she is wrong and now Mary is upset with you for not sticking by her side.
Obviously the example is to an extreme, but I think you get the gist.
I don't think you should take the other person's side no matter what. Try to distract "Mary" or derail the conversation if possible.
A loyal significant other would want to save their partner from embarrassing themselves, not add to their embarrassment.