  In this work, we present a collection of three-dimensional higher-order
symmetry protected topological phases (HOSPTs) with gapless hinge modes that
exist only in strongly interacting systems subject to subsystem symmetry
constraints. We use a coupled wire construction to generate three families of
microscopic lattice models: insulators with helical hinge modes,
superconductors with chiral Majorana hinge modes, and fractionalized insulators
with helical hinge modes that carry fractional charge. In particular, these
HOSPTs do not require spatial symmetry protection, but are instead protected by
subsystem symmetries, and support "fractonic" quasiparticle excitations that
move within only a low-dimensional sub-manifold of the system. We analyze the
anomaly structure for the boundary theory and the entanglement Hamiltonian, and
show that the side surfaces of these HOSPTs, despite being partially gapped,
exhibit symmetry anomalies, and can only be realized as the boundary of
three-dimensional HOSPT phases.
