Multi-phase flows occur in industrial applications such as sealing rings in internal combustion engines, electrolysis, foam casting, and many others. Simulation of these processes is a valuable tool to investigate relevant physical phenomena and perform design optimization in the early stages of component development. This is especially relevant when building multiple prototypes is more expensive than running multiple simulations. When the fluid domain deforms over time, additional challenges arise for modeling approaches and simulation software. Adaptive re-meshing can make simulations more efficient when facing a deforming fluid domain as well as moving inter-fluid boundaries typical in multi-phase flow. We investigate two versions of adaptive refinement for compressible/incompressible multi-phase flow with the level-set method using space-time finite elements [1] for a fixed domain benchmark problem and a newly introduced deforming domain test case. Both the accuracy and potential gains in performance are investigated. The two derived refinement schemes agree well with the rising bubble benchmark case and give reasonable results for the studied moving domain test case. Both approaches increase performance compared to uniform refinement for the investigated cases. A heuristic for estimating the relative performance of the two approaches is derived for assembly-heavy computations.