  We show that the gapless boundary signatures - namely, chiral/helical hinge
modes or localized zero modes - of three-dimensional higher-order topological
insulators and superconductors with inversion symmetry can be gapped without
symmetry breaking upon the introduction of non-Abelian surface topological
order. In each case, the fractionalization pattern that appears on surface is
`anomalous' in the sense that it can be made consistent with symmetry only on
the surface of a three dimensional higher-order insulator/superconductor. Our
results show that the interacting manifestation of higher-order topology is the
appearance of `anomalous gapped boundaries' between distinct topological orders
whose quasiparticles are related by inversion, possibly in conjunction with
other protecting symmetries such as TRS and charge conservation.
