It is usually said that anyon statistics are not well-defined if the anyons are massless. If I understand it correctly, the intuition is that any braiding has a finite duration, which means that any such process can always excite arbitarily low $\omega$-modes (if these exist, which is the case for gapless systems), ruining the necessary assumption of adiabaticity.
But what does that for example imply for a system of fermions with a Fermi surface? Are we not allowed to call the excitations 'fermions'?... [Or perhaps we can think even more fundamentally about the massless fermions in our universe.] Surely the fermionic nature must have particular effects, even if 'finite-time braiding' is not terribly well-defined. And if one agrees with that, why are we much more hesitant when it comes to more exotic anyons?