Have you considered increasing the bat population? An extra couple of hundred flying mice might be a nice ecologically friendly answer. You could also think about importing other avian species. Perhaps more pigeons?
I love bats. Problem is, bats are nocturnal and cluster flies are diurnal. Additionally, the seasons are off - the bats aren't here when the flies come out in the spring and are mostly gone by the time they swarm and cluster in the fall. Also, most bats tend to hunt in the air anyway, and these cluster flies spend almost all of their time on the ground or on the sunny walls of wooden structures.
More birds are logical, but interestingly bird populations have exploded here in recent years, I suspect partly in response to this overabundant food source. It's tricky hunting sometimes for them. The robins and starlings (an imported species themselves) tend to feast on them on the fields as the flies seek out worms to host their larvae, but later in the day when they cluster on walls the chickadees actually manage to hover and pick them off the walls, though obviously with great effort. We get a lot of humming birds in the summer, but they only eat small insects, the cluster flies are a little too large for them.
The other ideal predator (well not so ideal I guess) is also nocturnal: moths. We get some pretty large ones around here. I've actually mistaken them for bats more than once as their hunting behaviours are quite similar.
The one diurnal predator that is viable is ladybugs, and they cycle in population with the flies, but just don't seem to make a dent in them. Introducing additional ladybugs might help... but they exhibit many of the same annoying behaviours and they bite! (Cluster flies do not)
Then there's spiders. But again there's just way more flies than they could ever eat. And they do eat them in great quantities. It is amazing that spider eggs survive the winter here, but enough do that we still have an abundance of them.
or do it like they do with roaches. A pesticide that spreads when they come into contact with each other. The more they come into contact with each other the more it spreads.
That makes a lot of sense... as they get their name from the way they cluster in large groups.
I have even considered ways of turning them into a resource. Immense amounts of protein go to waste each year with these beasts. I've wondered if they might make good fish feed for aquaculture or perhaps dog food... If I could make a buck on these suckers I'd be rich!