Basic questions about fusion of two anyons

Suppose we have two anyons $a$ and $b$ on a manifold, and we use $|a\otimes b\rangle$ to label the corresponding wavefunction. Based on the fusion rule:

$a\otimes b=\oplus_c N_{ab}^c c$,

we may write the wavefunction $|a\otimes b\rangle$ as

$|a\otimes b\rangle=\oplus_c \psi_{ab}^c|c\rangle$,

or in terms of density matrix

$\rho_{a\otimes b}=\oplus_c|\psi_{ab}^c|^2\rho_c$,

where $\rho_c=|c\rangle\langle c|$.

Based on the literature such as the work by Kitaev and Preskill: http://arxiv.org/abs/hep-th/0510092, I know that [see Eq.(10) therein]

$|\psi_{ab}^c|^2=P_{ab\to c}=\frac{d_c}{d_ad_b}N_{ab}^c$,

where $d_i$ is the quantum dimension of anyon $i$. Consider the simplest case $N_{ab}^c=1$ or $0$, then $\psi_{ab}^c$ may be written as

$\psi_{ab}^c=\sqrt{\frac{d_c}{d_ad_b}}N_{ab}^c$,

up to a phase $e^{i\phi_{ab}^c}$ which is unknown here. My question is, what is the phase of $\psi_{ab}^c$?

In the above example, I understand that different topological sectors $c$ do not talk to each other, and thus this phase does not play an important role. But this phase may have some effects in other possible cases. Does anyone know how to deal with the phase $e^{i\phi_{ab}^c}$ here? Is there any good reference talking about this issue? Thanks very much!