an Abelian complex statistical phase from exchanging non-Abelian anyons?

Question

We have some discussions in Phys.SE. about the braiding statistics of anyons from a Non-Abelian Chern-Simon theory, or non-Abelian anyons in general.

May I ask: under what (physical or mathematical) conditions, when we exchange non-Abelian anyons in 2+1D, or full winding a non-Abelian anyon to another sets of non-Abelian anyons of a system, the full wave function of the system only obtain a complex phase, i.e. only $\exp[i\theta]$ gained (instead of a braiding matrix)?

Your answer on the conditions can be freely formulated in either physical or mathematical statements. This may be a pretty silly question, but I wonder whether this conditions have any significant meaning... Could this have anyon-basis dependence or anyon-basis independence. Or is there a subset or subgroup or sub-category concept inside the full sets of anyons implied by the conditions.

Answer 1

If you put a non-Abelian anyon and its anti particle on a sphere, then moving the non-Abelian anyon around its anti particle only induces an Abelian phase.

Also, twisting a non-Abelian anyon by 360$^\circ$ only induces an Abelian phase as well, which define the (fractional) spin of the non-Abelian anyon.