In causal perturbation theory there is no Higgs mechanism, unless one counts as it the need to introduce a scalar field to produce a causal description for a vector boson.
In traditional perturbation theory if you start with an action that has the Higgs field replaced by one shifted by a constant, you get directly the broken theory.
Thus perturbatively, the label ''broken symmetry'' is just a verbal convention.
Nonperturbatively, the situation may well be different since which symmetries are broken may depend on the boundary conditions considered (energy scale and chemical potential). Different boundary conditions lead to different asymptotic quasiparticle states. Causal perturbation theory almost exclusively considers only the vacuum sector, where this difference does not show up.