  The scalar potential of a multi-Higgs model can possess a rich structure of
minima and saddle points, which evolves in an intricate way as the parameters
change. In the hot early Universe, it could trigger multi-step phase
transitions, with exotic intermediate phases and peculiar domain wall
configurations. Here, we provide a glimpse into this richness with the example
of the three-Higgs-doublet model with the symmetry group $\Sigma(36)$, either
exact or softly broken. We present its phase diagram tracking not only the
global minimum but all of its extrema. In particular, we reveal parameter space
regions in which the deepest saddle point is charge breaking. This naturally
leads to phase transitions between neutral vacua which involve expanding and
colliding charge-breaking bubble walls. We also comment on opportunities of
multi-step phase transitions, on charge-breaking intermediate phases, and on
phase transitions between different charge-breaking vacua. We illustrate the
general discussion with two benchmark models, one of which possesses competing
saddle points and allows for emergence of bubbles of the same true vacuum but
with bubble walls of different nature. The intriguing cosmological implications
of all these possibilities deserve dedicated study.
