Lasagna garden it! Take out the tarp. Cover with cardboard, as much as you can. Then pile as much organic matter as you can on top of that (leaves, grass clippings, hay that doesn't contain seeds, etc--shoot for a foot if you can), then just a thin layer of compost or top soil (1-2 inches) and plant right into the top layer. Let nature decompose the thick grass mat over time, and your plant roots will just grow down, and the earthworms will be so happy.
For extra fun, watch Under Cover Farmer on youtube. See the last post in my journal for more on no-till farming, which I am obsessed with getting to work at my farm. Good luck!!