Wouldn't it work to have two end lines of some length set up with prussik's too? That the endlines are longer than needed to allow for adjustability and thicker trees, the tarp loops attached with prussiks, and the excess of the lines is hanging down from the tarp? I have so far only used a rl running above the tarp, through the loops on top. But this is a waste of rope as when tensioned hte tarp doesn't use the middle for support. It allows for another rope underneath to hang stuff from - one rope, one job as I recall from sailing.
Running it under the tarp would not work well for me (in theory): if it's used to support the tarp, you kind of lose the point if you hang stuff from it and it starts drooping, no? Also I worry about chafing when the wind picks up - is that not a concern?