  The Majorana lattice gauge theory purely composed of Majorana fermions on
square lattice is studied throughly. The ground state is obtained exactly and
exhibits the coexistence of symmetry breaking and topological order. The $Z_2$
symmetry breaking of matter fields leads to the intertwined antiferromagnetic
spin order and $\eta$-pairing order. The topological order is reflected in the
$Z_2$ quantum spin liquid ground state of gauge fields. The Majorana lattice
gauge theory, alternatively can be viewed as interacting Majorana fermion
model, is possibly realized on a Majorana-zero-mode lattice.
