Thought of this in the middle of the night!
Does a PV system produce any output during the night with a full moon?
Is there enough reflected sunlight?

Yes, but it is usually less than the loss in the system. I have messed with ours and doing a direct connect with a full moon at the right angle our system was making about 30 watts, during a sunny day our system makes about 2000w. The solar charge controller won't "wake up" until the voltage rises enough and it never hits that in a full moon, but just throwing an amp meter between the panels and batteries I saw the 30w. On a completely overcast day the system will generate about 200w, snow is the worst shutting them completely down :)

I love measuremnets, but I suspect this one was incorrect somehow....the luminous flux of the full moon is 10^-5 that of the full sun. A 2KW array would make ~20 milliwatts maximum (if it could run at the same eff). Perhaps the 20W was leakage current from the batteries flowing back through the panels?