So I un-plyied some 20 cordonnet thread, which I assumed was more like braid, but its 6 ply spun as three 2ply strands are plyed into one thread. I tried it out, plying three lengths of yarn and then plying them together.
I tatted a ring with it and its strong and isn't too fuzzy. It worked great. I over spun the 2 plys to make sure it held together during the last ply.
Now, I have to consider spinning very fine singles to bring the finished size of the yarn down, though I do want it to be thicker that 20 cordonnet. I have some design ideas that will use thicker thread.