Liquid-gas flows that involve compressibility effects occur in many engineering contexts, and high-fidelity simulations can unlock further insights and developments. Introducing several numerical innovations, this work details a collocated, volume-of-fluid, finite-volume flow solver that is robust, conservative, and capable of simulating flows with shocks, liquid-gas interfaces, and turbulence. A novel hybrid advection scheme, which combines an unsplit semi-Lagrangian method with a centered method, provides stability while minimizing dissipation. A pressure projection scheme makes multiphase compressible simulations much less costly, and formulating this projection as incremental reduces numerical dissipation further. Local relaxation to mechanical equilibrium is used to properly solve for the pressure and energy fields in multiphase contexts. The complete algorithm is validated with benchmark tests in one, two, and three dimensions that evaluate the accuracy and stability of the approach in predicting compressible effects, turbulent dissipation, interface dynamics, and more through comparisons with theory, experiments, and reference simulations. Finally, the numerical approach is applied in two detailed studies of atomization. The first involves a liquid jet, at multiple flowrates, in a Mach 2 crossflow, and the second explores the effect of strong acoustic fields on a turbulent liquid jet traveling through a periodic domain.