Ive had nothing but trouble with tarps every time i have used them to cover one thing or another. The best tarps i ever used are the kind used by truckers to cover lumber etc. on the flatbeds. A truck stop or truck supply place will be your best bet on finding them. They are the longest lasting tarps ive ever found.

Consider the Billboard Vinyl.
Used.
Generally free.
From large advertising billboard companies, like (evil! evil!) Clear Channel.
You'd want to get in touch with the yard boss and ask him if during his next yard purge of obsolete billboard vinyls, if you can swing by in your truck (and yes, it HAS to be a truck! these things can get heavy!) and get a Jumbo billboard vinyl?
(Jumbos' generally run 20' x 60' or so and weigh several hundred pounds.)
Bring him a case of beer for his workers, for when the day is over. Makes him popular and costs you almost nothing.

btw: Billboard vinyls have synthetic fibers running through the vinyl and are thicker than some pond liners I've seen, and they've usually finished off-gassing by the time their ad campaign has run out.
I love them and wish I could get more.

If you haven't already taken a look at it, I would suggest tyvek.

en.wikipedia.org/wiki/Tyvek

It is almost as light as paper, extremely tear resistant, won't break down under UV, and it's totally waterproof. Ultralight backpackers swear by this stuff.

Best of all you can pick it up at HD or Lowes for pretty damned cheap. If memory serves you can get a roll (4' x 50') for something like $120.

Have you checked out billboard vinyl? It is the cheapest and best tarp available. Sun resistance is great.

Might be the same as billboard vinyl, but when i was in the signage industry we used a material called flexface. was a vinyl impregnated mesh material used for outdoor signage. PVC is durable as it gets. as for offgassing, of course. plastic will do this for a term and then come close to stabilizing. used materials could theoretically be more stable?

www.tarpsplus.com/greytarps.html