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In many BSM theories, especially GUTs, introducing a Brout–Englert–Higgs effect allows for multiple breaking patterns of the gauge symmetry. The possibility to select a particular pattern is usually decisive for the phenomenological viability of a theory. Beyond perturbation theory it is necessary to replace the Brout–Englert–Higgs effect by a manifestly gauge-invariant description. We study the simplest case with multiple breaking patterns, an SU (3) Yang–Mills theory coupled to a single scalar 'Higgs' field in the adjoint representation, on the lattice. We find that only one pattern remains at fixed parameters and gauge-fixing strategy, and that the associated quantum effective potential emerges from a non-trivial interplay of many aspects. [ABSTRACT FROM AUTHOR]