Are anyons non local?

Studying anyonic statistics in 2 dimensions, I naturally thought to ask the question of whether anyons are non local, since as we braid one around another, no matter the distance between the two, one must feel the presence of the other.

I understand that anyons result at the end of extended string excitations. What exactly does it imply?

As a corollary, fermions have the same sort of exchange statistics, but they are definitely local. They are also attached to Wilson lines to make them gauge invariant.

So to sum it up, are these particles non-local? Does this have anything to do with the gauge strings at are attached to the particles to make them gauge invariant? I suspect it does since mathematically the phase is only introduced when the two strings of the two objects cross.