This idiom 'run with the hare and hunt with the hound' means figuratively to support both sides of a dispute.
Here it refers to 2 persons or parties, but it could also refer in other contexts to supporting 2 competing sides in an argument.
The idiom shows how difficult and quite impossible it is to do both at the same time, since literally you can't be on both sides hunting the hound and running with the hare.