Thin cork handlebar tape/wrap?
 

I recently parted out a bike which had thin cork tape. It had no adhesive on the back, so it came off whole. I might see if I can reuse it.

It reminded me of how nice and thin cork tape used to be. Does anyone still produce thin cork tape?

I bought a bike and the seller gave me some cheap Bell bar tape he didn't use. Probably got it at walmart. I just put it on a bike and it's actually very nice. Looks like Profile tape but a tad thinner. It does have adhesive but I've never had a problem removing that kind. If it gets too sticky just use a heat gun on it so you don't tear it up.