We had two flat tires. One was a sidewall puncture, most likely caused by a small stick (remember, bushes in the desert grow very slowly and develop an incredibly hard wood - as hard as nails) we fixed it with a Safety Seal plug.
The other tire damage was more serious.
It was compression cut, caused by driving (too fast) over a rock as depicted below. The tire sidewall is cut between the rock and the rim. This usually happens when you try to avoid a rock (but don't succeed) and hit it only with a small portion of the tread. Most of those damages occur on one of the right tires - in our case it was the front right tire. There is no field fix for this kind of damage.
These accidents happen because it is not easy to spot rocks that could endanger our tires. Rocks like the ones you see below are "icebergs" - only 10% of their body shows. The only warning sign is that they are somewhat blackened - blackened from tires that hit before.
Don't try to avoid them. You might fail. Drive slowly with the center of your tire(s) over them to avoid compression cuts. See how