How exasperating, that is indeed a problem with sou but I presume you mean it contains actual roots – if only seeds they are easy to hoe.
But roots of perennial weeds, no that is not right, ask for your money back at least (I guess it is sold as a mix for garden use?).
Meanwhile I wonder of the tarp is totally excluding light because two years should have killed or at least severely weakened them. Here, just a year in total darkness sees bindweed weakened to the point that occasional use of a trowel becomes an easy way to control it.
I do not have experience of your two weeds but doubt they can be as indestructible as you are finding. Look through the tarp at the sun and see how light it is.
