There are nylon tarps available that aren't as light as the SilNylon tarps. Check out the tarps at Campmor.

The lightweight tarps are "1.1 oz. silicone impregnated ripstop nylon. Very strong and extremely water resistant." I carry the small 6x8 in my get home bag (9.2 oz), it packs small. For comparison, the 10x12 weighs 19 oz.

Next up would be the Coated Nylon Tarps which are "1.9 oz. per square yard reinforced high count taffeta nylon. Very light and flexible. Waterproof urethane coating." I keep a 12x16 in my truck, it's a nice piece of gear to have in a pinch. 10x12 weighs 37 oz.

Last would be the Ripstop Woven Polyethylene Tarps. Definitely less spendy, but they're heavier. I've used them in the past, but prefer nylon. 10x12 weighs 4 lbs (64 oz).

If you don't intend to carry one on your back, the Polyethylene Tarps might work fine.