I agree, I hate those consumer blue tarps aswell.
Not a permaculture answer, but back when we had a public landfill, I managed to find what I believed to be small 60x40 silage tarps. They were black/grey and are rather heavy. I now use them to stunt grass before planting an area and actually have never stored one of them - it's been outside 2 whole years now. Besides the initial holes, the reason it was likely thrown away, I haven't noticed any ripping and I drag it over gravel a lot. But I've never used it on a brushpile either.
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You could probably find some canvas though and use that. Some people I know had sweatlogdes and 2 layers of canvas kept them dry even during 1 inch rains. Something like
this might fit the bill.