  Theories with more than one scalar field often exhibit phase transitions
producing potentially detectable gravitational wave (GW) signal. In this work
we study the semi-annihilating $\mathbb{Z}_3$ dark matter model, whose dark
sector comprises an inert doublet and a complex singlet, and assess its
prospects in future GW detectors. Without imposing limits from requirement of
providing a viable dark matter candidate, i.e. taking into account only other
experimental and theoretical constraints, we find that the first order phase
transition in this model can be strong enough to lead to a detectable signal.
However, direct detection and the dark matter thermal relic density constraint
calculated with the state-of-the-art method including the impact of early
kinetic decoupling, very strongly limit the parameter space of the model
explaining all of dark matter and providing observable GW peak amplitude.
Extending the analysis to underabundant dark matter thus reveals region with
detectable GWs from a single-step or multi-step phase transition.
