I'm so sorry you're having to deal with this mess! My advice, for what it's worth, is to be preemptive. Make sure your birds have no contact with wild birds of any sort.
Cover your run. This could be anything from a solidly anchored tarp to a permanent structural roof. The key is to prevent any "deposits" from overhead. (Bonus: you can stay out of the weather when out with your birds!)
Replace any large-opening wire with hardware cloth no bigger than an inch. That prevents smaller birds (like songbirds) from entering your set-up. This also becomes an early-warning system, of sorts. If a stray bird gets in, you have a hole to find. A hole big enough to let in a sparrow can also let in snakes and other small predators.