We study gravitational wave (GW) production in strongly supercooled cosmological phase transitions, taking particular care of models featuring a complex scalar field with a U$(1)$ symmetric potential. We perform lattice simulations of two-bubble collisions to properly model the scalar field gradients, and compute the GW spectrum sourced by them using the thin-wall approximation in many-bubble simulations. We find that in the U$(1)$ symmetric case the low-frequency spectrum is $proptoomega$ whereas for a real scalar field it is $proptoomega^3$. In both cases the spectrum decays as $omega^{-2}$ at high frequencies.
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