I agree with Lori--that's polyurethane. It's what makes the tent waterproof. It's obviously a waterproof coating, and most silnylon is impregnated, rather than coated, with silicone. Most standard tent manufacturers (such as Sierra Designs) use polyurethane coated nylon. Without the coating, the tent will leak like a sieve.

IMHO, you have a choice--either the procedure Lori quotes or the garbage can.

I have this suspicion that the tents were rather aged and ready to delaminate when you got them. The washing just hastened the process.