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In this answer, it is said that gauging the symmetry which protects a symmetry-protected topological (trivial) phase gives something "morally very similar" to a phase with a topological order. What are the properties which the resultant phase generically does and does not share with the topologically ordered phase?
Any reference which discusses the issue systematically would be appreciated.
I think that when we gauge a global symmetry of an SPT phase, we get an honest TO phase and not just a moral one. The gauge fluxes act as quasiparticles of the TO phase, and the quasiparticle content and statistics gives us invariants of the TO and hence of the "parent" SPT phase. There are many details of this that I am not expert in though, and I'd be happy to be corrected.
Here are some references:
Levin and Gu, Braiding statistics approach to symmetry-protected topological phases, https://arxiv.org/abs/1202.3120
Wen, Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions, https://arxiv.org/abs/1301.7675
Hung and Wen, Universal symmetry-protected topological invariants for symmetry-protected topological states, https://arxiv.org/abs/1311.5539
