We developed a multi-pronged approach. Fly problems occur here when rainy season starts in June. I rake out runs every week, bag what I rake, and use it for fertilizer. It's weak enough to side dress on mature plants, and any excess goes into the compost piles. The coops have deep litter. I use agricultural lime in the runs, all of which have steel paneled roofs to keep most of the rain out. And put up fly bags if we start to have an infestation. The girls are accomplished fly catchers, and do a terrific mop-up job on any fly that happens to wander in.
I will say that it's a challenge when we have any tropical systems. Some seasons are worse than others, and even a weak hurricane passing by will drench everything. One thing I have learned is to avoid using any kind of hay, or chips, in the runs, due to our soil composition. That's how they made bricks back in the old days. The ground will get rock hard, absorb nothing, and turn into fly-heaven.