  We consider fermionic systems in which fermion parity is conserved within
rigid subsystems, and describe an explicit procedure for gauging such subsystem
fermion parity symmetries to obtain bosonic spin Hamiltonians. We show that
gauging planar or fractal subsystem fermion parity symmetry in three spatial
dimensions gives rise to a plethora of exactly solvable spin models exhibiting
novel gapped fractonic orders characterized by emergent fermionic gauge theory.
The low energy excitations of these models include fractional quasiparticles
with constrained mobility and emergent fermionic statistics. We illustrate this
phenomenon through a series of examples including fermionic analogs of both
foliated fracton phases and fractal spin liquids. We find that the foliated
analogs actually exhibit the same fractonic order as their bosonic
counterparts, while this is not generally the case for fermionic fractal spin
liquids.
