  The hallmark of a 2 dimensional topologically ordered phase is the existence
of deconfined `anyon' excitations that have exotic braiding and exchange
statistics, different from those of ordinary bosons or fermions. As opposed to
conventional Landau-Ginzburg-Wilson phases, which are classified on the basis
of the spontaneous breaking of an underlying symmetry, topologically ordered
phases, such as those occurring in the fractional quantum Hall effect, are
absolutely stable, not requiring any such symmetry. Recently, though, it has
been realized that symmetries, which may still be present in such systems, can
give rise to a host of new, distinct, many-body phases, all of which share the
same underlying topological order. A useful approach to classifying SETs is to
determine all possible such symmetry actions on the anyons that are
algebraically consistent with the anyons' statistics. Remarkably, however,
there exist symmetry actions that, despite being algebraically consistent,
cannot be realized in any physical system, and hence do not lead to valid 2d
SETs. One class of such `anomalous' SETs, characterized by certain disallowed
symmetry fractionalization patters, finds a physical interpretation as an
allowed surface state of certain 3d short-range entangled phases, but another,
characterized by some seemingly valid but anomalous permutation actions of the
symmetry on the anyons, has so far eluded a physical interpretation. In this
work, we find a physical realization for these anomalously permuting SETs as
surface theories of certain 3d long-range entangled phases, completing our
understanding of general anomalous SETs in 2 dimensions.
