I recall seeing a thread a while back that indicated the Spiderm Anneeyys guy has gone out of his way to avoid sharing the source of his red fabric. Ordinarily I'd say that runs counter to the spirit of this community, but the fact is that he's able to command a pretty high premium for his work so I can't entirely fault him for wanting to maintain an air of exclusivity. I can only assume that the fabric is hard (and expensive) to come by, anyway, considering that a thorough online search hits a dead end when trying to find a comparable fabric. Virtually all "satin/charmeuse" stretch fabric has a fairly low lycra/spandex content so it wouldn't be all that practical for a Spidey costume (unless it was carefully tailored to fit a specific person).
There are tons of red spandex out there, but none really match that holy grail of sheen and contrast. At least none that I've found.