  We consider first order cosmological phase transitions (PT) happening at late
times, below Standard Model (SM) temperatures $T_{\rm PT} \lesssim$ GeV. The
inherently stochastic nature of bubble nucleation and the finite number of
bubbles associated with a late-time PT lead to superhorizon fluctuations in the
PT completion time. We compute how such fluctuations eventually source
curvature fluctuations with universal properties, independent of the
microphysics of the PT dynamics. Using Cosmic Microwave Background (CMB) and
Large Scale Structure (LSS) measurements, we constrain the energy released in a
dark-sector PT. For 0.1 eV $\lesssim T_{\rm PT} \lesssim$ keV this constraint
is stronger than both the current bound from additional neutrino species
$\Delta N_{\rm eff}$, and in some cases, even CMB-S4 projections. Future
measurements of CMB spectral distortions and pulsar timing arrays will also
provide competitive sensitivity for keV $\lesssim T_{\rm PT} \lesssim$ GeV.
