  We study production of gravitational waves (GWs) in strongly supercooled
cosmological phase transitions, considering classically conformal scalar
electrodynamics as a benchmark model. We perform lattice simulations in order
to study the evolution of the complex scalar field and the gauge field in
two-bubble collisions. From these simulations we extract the scaling of the GW
source in the collision, which we then use in many-bubble simulations in the
thin-wall limit to calculate the resulting GW spectrum. We find that in
presence of the gauge field the field gradients sourcing GWs decay with the
bubble size as $\propto R^{-3}$ similarly to the real scalar case. This leads
to a GW spectrum that follows $\Omega_{\rm GW} \propto \omega^{2.3}$ at low
frequencies and $\Omega_{\rm GW} \propto \omega^{-2.9}$ at high frequencies. We
also explore the spectra resulting for different decay laws finding that the
envelope result predicting $\omega^{-1}$ behaviour at high frequencies does not
provide an accurate description of the spectrum in realistic models.
