. . . Poop all over the carpet. Actually, not all over but in tiny discrete piles, as if his sphincter muscles are unable to close off.
I stepped in it again this morning and I need help.
What is causing this? What can I do about it?
IMO, your cat may have a very dangerous illness - I will not even speculate what, as there are several that can lead to this sort of behavior. Take your poor kitty to the vet! And good luck with kitty! 
What could be wrong with him?
I know that, in general, his poop smells extremely sulfurous [sp?]. Also, when we first got him, he had chronic diarrhea. We did take him to the vet that time and they said he had a bacterial infection, for which we gave him antibiotics.
We feed him Prescription Diet Feline I/D dry cat food and have a Litter Maid so his box is always clean. He doesn’t eat table scraps.
I concur, make a vet appointment asap. That’s not something you want to wait on…even if its just diarrhea, kitty could get dehydrated pretty quickly.
He doesn’t really have diarrhea any more. It’s just that his anus apparently doesn’t close up tight enough after he’s done using his box to prevent excess poop from seeping out when he’s walking around the apartment.