Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation Hilbert space of $c$ to the one of $a \otimes b$ and assume that $U$ is a partial isometry. By Schur's lemma $U$ is determined up to a phase factor.
Question: Can we assume that $U$ has real (Clebsch-Gordan) coefficients?
If not, is there a good class of examples for which this is true? Or is it true for free orthogonal/unitary QG's?
