  We study the thermal phase transitions of a generic real scalar field,
without a $Z_2$-symmetry, referred to variously as an inert, sterile or singlet
scalar, or $\phi^3+\phi^4$ theory. Such a scalar field arises in a wide range
of models, including as the inflaton, or as a portal to the dark sector. At
high temperatures, we perform dimensional reduction, matching to an effective
theory in three dimensions, which we then study both perturbatively to
three-loop order and on the lattice. For strong first-order transitions, with
large tree-level cubic couplings, our lattice Monte-Carlo simulations agree
with perturbation theory within error. However, as the size of the cubic
coupling decreases, relative to the quartic coupling, perturbation theory
becomes less and less reliable, breaking down completely in the approach to the
$Z_2$-symmetric limit, in which the transition is of second order.
Notwithstanding, the renormalisation group is shown to significantly extend the
validity of perturbation theory. Throughout, our calculations are made as
explicit as possible so that this article may serve as a guide for similar
calculations in other theories.
