  Simple scalar-singlet extensions of the Standard Model with a (spontaneously
broken) $Z_2$ symmetry allow for a strong first order electroweak phase
transition, as sought in order to realize electroweak baryogenesis. However
they generically also lead to the emergence of phenomenologically problematic
domain walls. Here we present a framework with a real scalar singlet that
features a different thermal history that avoids this problem by never
restoring the $Z_2$ symmetry in the early universe. This is accomplished by
considering $D>4$ operators that emerge on general grounds, understanding the
model as the low energy tail of a more complete theory, like for example in
composite Higgs scenarios. Sticking to the latter framework, we present a
concrete $SO(6)/SO(5)$ composite realization of the idea. To this end, we
additionally provide a complete classification of the structure of the Higgs
potential (and the Yukawa couplings) in $SO(6)/SO(5)$ models with fermions in
the ${\bf 1, 6, 15}$ or ${\bf 20^\prime}$ of $SO(6)$.
