A spin structure on a Riemannian bundle of rank >2 is the lift of the structure group from $\text{SO}(n)$ to its universal cover $\text{Spin}(n).$ It may also be defined in the case $n=2$ as the lift of the structure group to a double cover of $\text{SO}(2)$, which is of course not the universal cover.

So what about lifts to other covers of $\text{SO}(2)$. Does the lift to a three-to-one cover of $\text{SO}(2)$ have a topological obstruction living in $H^2(B,\mathbb{Z}/3)$? Is the concept of a lift to the universal cover of $\text{SO}(2)$ affected by the unusual statistics of anyons?

Searching the literature for spin structures, I found only the double cover case. And searching the archive for anyons yields nothing but solid state physics articles. So I would be happy just to have a reference to any work discussing this case.