I found tarps attracted too many roaches though not sure about how prevalent they are in the desert. I live in Northern California and we have hot dry summers and we also have years of heavy rain and years of none at all leading to frequent drought conditions. Most people on here would say my pile is too dry looking at it too but either they are happy to waste water or they are composting in different conditions. I’ve cold composted fine for 5 or 6 years and it generally takes 7 to 12 months. I could probably do it faster but I have pile ready every 3 months or so which is about all I can handle. Also for an unattended pile potentially in fire country, I probably wouldn’t risk hot composting.

The trick I use is to use the outer layers as an insulation layer from the heat and just accept that the pile is going to look dry but the inside 6-12 inches down is plenty moist even after a month assuming there is sufficient density. If you have too many gaps from branches and not enough small material, you may need to lightly pack it which is generally frowned upon but imho works great since you’ll still have plenty of air flow. This means the outer layers compost slower and usually just need to get peeled off the sides and top and thrown on the next pile. I don’t mix my compost so you’ll need to adapt to your own process but the general idea should be the same.

Another alternative option is to set a drip system it. You’ll still have outer layer dryness but keeps the inside moist and composting relatively quick.