Excluding posts with only one specific category

I have what I believe to be a somewhat obscure need. I have a category I wish to exclude from the loop – which is easy. If the post has multiple categories, however, it needs to be included.

I have searched and searched and found ways to exclude categories and multiple categories, but I haven’t found a way to exclude posts that exist only in one specific category but include them if they’re in the specific category and any other category.

I’m sure it’s just a quick tweak of the query, but I can’t find it.