We are in Scottsdale and we have the same issue. We have 4 orange trees, a couple plums, two grapefruit, one lemon, one key lime, a pomegranate and a cocktail tree. The only things they don't eat are the lemons and limes. Everything else is gone before it is ripe. They eat through everything, including the stucco. I don't leave food out at night, and I have it set up so there is very little waste, I have trays under the girl's feeders and they catch all but a sprinkle of food. I also pull water at night. They eat the citrus and the pomegranates. I can't cut every tree down. I have zap traps that occasionally work, but we are in a livestock area, surrounded by horse properties so the rats live there too. This year they have eaten through the stucco and roof areas and now we have herds of them in the roof. Like stampedes. We have paid "professionals" thousands to seal the roof only to have them eat through in several other areas. The big issue we have is that once we seal the holes they die in the attic and it smells for a few weeks, until they eat in/out. To be clear, it's been an issue before we had chickens. My point in the post is that in PHX it's not really about the chickens, roof rats are bad because of the citrus/fruit trees. Chicken feed is just a sweet bonus for them.
I have the runs and coops set up so there is an automatic door on each of the coops. They can't get into the girls at night. I have two coops, one Omlet with an auto door, one prefab with an omlet auto door. All the prefab coop panels are double layered hardware cloth. The rats have the run, but they can't access the coops. I have hardware clothed a small run, but they find ways in. Always. The coops are the only thing I have reliably been able to secure. In the winter the zap traps do pretty well. I also had great success with live trapping but I don't have the heart to kill them and driving them into the desert isn't sustainable.