  A generalization of Wilsonian lattice gauge theory may be obtained by
considering the possible self-adjoint extensions of the electric field operator
in the Hamiltonian formalism. In the special case of 3D $\mathrm{U}(1)$ gauge
theory these are parametrised by a phase $\theta$, and the ordinary Wilson
theory is recovered for $\theta=0$. We consider the case $\theta=\pi$, which,
upon dualization, turns into a theory of staggered integer and half-integer
height variables. We investigate order parameters for the breaking of the
relevant symmetries, and thus study the phase diagram of the theory, which
shows evidence of a broken $\mathbb{Z}_2$ symmetry in the continuum limit, in
contrast to the ordinary theory.
