  The description of electroweak physics using perturbation theory is highly
successful. Though not obvious, this is due to a subtle field-theoretical
effect, the Fr\"ohlich-Morchio-Strocchi mechanism, which links the physical
spectrum to that of the elementary particles. This works because of the special
structure of the standard model, and it is not a priori clear whether it works
for structurally different theories. Candidates for conflicts are, e.g., grand
unified theories. We study this situation in a toy model, a $SU(3)$ gauge
theory with two Higgs fields and a breaking pattern $SU(3) \rightarrow SU(2)
\rightarrow 1$. This mimics the weak-Higgs sector of the standard model. We
determine the leading order predictions for the gauge invariant spectrum in
this theory, and discuss a setup to test them using lattice gauge theory.
