This guy says that a sticky fly or tarp can be mitigated and he gives some suggestions:
Having a sticky tent or rain fly can be quite frustrating to set up. Whether you are working on your own or with help of a fellow camper, having to pull apart the panels because they are tacky and stuck together is a pain. This makes the whole process a lot more complicated and time consuming...
laidbackcamping.com
Those old Serratus tarps were workhorses. I've got three that are likely the same vintage as yours and they are just starting to feel tacky. I get drops here and there, but they probably have a few years left in them.
My MEC Supertarp is newer (maybe 10 years old) but already it is tackier than the Serratus tarps. I think that is likely because it is so huge that it is hard to store loosely folded.
If you are looking for a replacement, I have several DD Hammocks 3x3 tarps that I love:
DD Tarp 3x3. Ultralight, tough and extremely versatile tarp with 19 attachment points.
www.ddhammocks.com
Cheers,
Andrew