If you don't have enough teams for AND to be a good user experience, NOT might be a better user experience.

Count matches, sort, and then put separators between different levels of matching.

So the top of the list would be companies with 8 matches, then there is a separator, then there are companies with 7 matches, then there is a separator, and so on... all the way down to zero matches. The list is always everything, but the sort order changes.

I figured that was why it was an OR right now too.