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.