What the heck is this stuff??
 

We have inherited my in-laws dining set and I have a question about the
seats. The cording on them is not rattan but rather a rolled paper. It
doesn't look like an easily performed craft because the weave piles up as it
approaches the center of the seat. Are any of you familiar with this sort
of seating? What is it called? Where would I look to have it redone? A
specialty shop? TIA for your help.
Chuck