  Extra CP-violating source for electroweak baryogenesis can dynamically appear
at finite temperature in the complex two-Higgs doublet model, which might help
to alleviate the strong constraints from the electric dipole moment
experiments. In this scenario, we study the detailed phase transition dynamics
and the corresponding gravitational wave signals in synergy with the collider
signals at future lepton colliders. For some parameter spaces, various phase
transition patterns can occur, such as the multi-step phase transition and
supercooling. Gravitational waves complementary to collider signals can help to
pin down the underlying phase transition dynamics or different phase transition
patterns.
