  Cosmological first order phase transitions are typically associated with
physics beyond the Standard Model, and thus of great theoretical and
observational interest. Models of phase transitions where the energy is mostly
converted to dark radiation can be constrained through limits on the dark
radiation energy density (parameterized by $\Delta N_{\rm eff}$). However, the
current constraint ($\Delta N_{\rm eff} < 0.3$) assumes the perturbations are
adiabatic. We point out that a broad class of non-thermal first order phase
transitions that start during inflation but do not complete until after
reheating leave a distinct imprint in the scalar field from bubble nucleation.
Dark radiation inherits the perturbation from the scalar field when the phase
transition completes, leading to large-scale isocurvature that would be
observable in the CMB. We perform a detailed calculation of the isocurvature
power spectrum and derive constraints on $\Delta N_{\rm eff}$ based on CMB+BAO
data. For a reheating temperature of $T_{\rm rh}$ and a nucleation temperature
$T_*$, the constraint is approximately $\Delta N_{\rm eff}\lesssim 10^{-5}
(T_*/T_{\rm rh})^{-4}$, which can be much stronger than the adiabatic result.
We also point out that since perturbations of dark radiation have a
non-Gaussian origin, searches for non-Gaussianity in the CMB could place a
stringent bound on $\Delta N_{\rm eff}$ as well.
