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We develop an exact map of all states and operators from 2D lattices ofspins-$1/2$ into lattices of fermions and bosons with mutual semionicstatistical interaction that goes beyond previous dualities of $\mathbb{Z}_2$lattice gauge theories because it does not rely on imposing local conservationlaws and captures the motion of ``charges'' and ``fluxes'' on equal footing.This map allows to explicitly compute the Berry phases for the transport offluxes in a large class of symmetry enriched topologically ordered states withemergent $\mathbb{Z}_2$ gauge fields that includes chiral, non-chiral, abelianor non-abelian, that can be perturbatively connected to models where the visonsare static and the emergent fermionic spinons have a non-interactingdispersion. The numerical complexity of computing such vison phases reducestherefore to computing overlaps of ground states of free-fermion Hamiltonians.Among other results, we establish numerically the conditions under which theMajorana-carrying flux excitation in Ising-Topologically-Ordered statesenriched by translations acquires $0$ or $\pi$ phase when moving around asingle plaquette.

Berry Phases of Vison Transport in $\mathbb{Z}_2$ Topologically Ordered States from Exact Fermion-Flux Lattice Dualities