Rep(Z7 ⋊ Z3) and Rep(Z5 ⋊ Z4) are the lowest-rank braided fusion categories known whose F-symbols are inherently complex, as proven by explicit non-real gauge-invariant quantities.
Unphysical diagonal modular invariants
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Anyons and Inherently Complex F-symbols
Rep(Z7 ⋊ Z3) and Rep(Z5 ⋊ Z4) are the lowest-rank braided fusion categories known whose F-symbols are inherently complex, as proven by explicit non-real gauge-invariant quantities.