I prefer a heavy weight shirt, too, so this definitely wasn't going to work.
OK, out with my favorite piece of equipment.....a large pair of shears...
Bingo! - I left the huge heap (to tidy up another day), because I realized I had what I needed already. I stitched the back and front of the tank together to make one thick layer, and used fabric from my stash to make a striped back.
In the striped fabric, I cut the same basic shape, adding a little flare through the body, then slit up the center back almost to the neck. I twisted the middle and sewed the center back together again.
I added a narrow black band at the front neckline, and I now have a tank that I'll actually wear!