Reality and Complexity of $F$-symbols in $2+1$d Topological Phases

The $F$-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the $F$-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all $F$-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated $F$-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,'' or equivalently when the associated "twisted'' Frobenius-Schur (or "generalized'' Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose $F$-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex $F$-symbols.