  For a tetragonal material, order parameters of $p_x$ and $p_y$ symmetry are
related by rotation and hence have the same $T_{\rm c}$ at a mean-field level.
This degeneracy can be lifted by a symmetry-breaking field, like (uniaxial)
in-plane strain, such that at $T_{\rm c}$, the order parameter is only of $p_x$
or $p_y$ symmetry. Only at a lower temperature also the respective other order
parameter condenses to form a chiral $p$-wave state. At the mean-field level,
the derivative of $T_{\rm c}$ with strain is discontinuous at zero strain. We
analyze consequences of (thermal) fluctuations on the strain-temperature phase
diagram within a Ginzburg-Landau approach. We find that the order-parameter
fluctuations can drive the transition to be weakly first order, rounding off
this discontinuity. We discuss the possibility of a second-order transition
into a non-superconducting time-reversal-symmetry-breaking phase and
consequences for the spin-triplet superconductor Sr$_2$RuO$_4$.
