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