Incorrect Output From 7805
Seems @JohnBirckhead has hit the nail on the head. 

You need to add a diode to the output of the LM7805 to ground. The datasheet suggests a 1N400x, but I would use something like a 1N5819 Schottky diode which will prevent almost all current from getting into the substrate diodes. 



The effect you are seeing is likely caused by the output terminal being pulled below ground (something that can't happen easily when you don't have a negative supply). Likely that shuts the regulator output down by some internal sneak paths through the substrate isolation diodes. 

I usually put diodes like the datasheet suggests into such supplies as a matter of course, even when they are sequenced up and down. Cheap insurance against strange happenings. Often you can use a small dual like BAT54S and protect both regulators. 

Incidentally, this behavior may well vary from one supplier to another of the 7805 chip. It's outside the normal range of operation. In a quick test, I was unable to duplicate the behavior OP is seeing even with -200mA applied to the output (Motorola part). OP's part number is apparently ST, which has this warning about "latch-up problems" in the datasheet: