  We study $\mathbb{Z}_2$ topologically ordered states enriched by
translational symmetry by employing a recently developed 2D bosonization
approach that implements an exact $\mathbb{Z}_2$ charge-flux attachment in the
lattice. Such states can display `weak symmetry breaking' of translations, in
which both the Hamiltonian and ground state remain fully translational
invariant but the symmetry is `broken' by its anyon quasi-particles, in the
sense that its action maps them into a different super-selection sector. We
demonstrate that this phenomenon occurs when the fermionic spinons form a weak
topological superconductor in the form of a 2D stack of 1D Kitaev wires,
leading to the amusing property that there is no local operator that can
transport the $\pi$-flux quasi-particle across a single Kitaev wire of fermonic
spinons without paying an energy gap in spite of the vacuum remaining fully
translational invariant. We explain why this phenomenon occurs hand-in-hand
with other previously identified peculiar features such as ground state
degeneracy dependence on the size of the torus and the appearance of dangling
boundary Majorana modes in certain $\mathbb{Z}_2$ topologically ordered states.
Moreover, by extending the $\mathbb{Z}_2$ charge-flux attachment to open
lattices and cylinders, we construct a plethora of exactly solvable models
providing an exact description of their dispersive Majorana gapless boundary
modes. We also review the $\mathbb{Z}\times (\mathbb{Z}_2)^3$ classification of
2D BdG Hamiltonians (Class D) enriched by translational symmetry and provide
arguments on its robust stability against interactions and self-averaging
disorder that preserves translational symmetry.
