Yah, the tape-wounds have that vibe. They still are basically a regular round-wound (with the last layer being flat nylon tape), so they still have 'that' sound.
You might try double-bass strings like
http://www.juststrings.com/lab-1006.html
...but if you think about it, a lot of double-bass strings still have steel wrapping (except like the above), and even if you go to cello strings it's the same. The nylon core with silver plate still has metal, though silver is not ferrous.
Anyhow, point is, if you're thinking about trying to get a true acoustic tone, or something similar, and are going to use piezo or optical, then...it shouldn't matter what the strings themselves are made of, i.e., double-basses with their normal metal core strings, miked or with piezo, still sounding like a double-bass. At that point it's about the design, materials and construction of the instrument itselff and not so much the string.
I was even looking at Gamba strings and it's pretty much the same thing; gut core with metal wrapping. I mean at a certain point, esp. with piezo pickups, it's about the sound of the wood of the body/neck/bridge and not so much about the strings themselves.
You can try it though.
