  We suggest an appealing strategy to probe a large class of scenarios beyond
the Standard Model simultaneously explaining the recent CDF II measurement of
the $W$ boson mass and predicting first-order phase transitions (FOPT) testable
in future gravitational-wave (GW) experiments. Our analysis deploys
measurements from the GW channels and high energy particle colliders. We
discuss this methodology focusing on the specific example provided by an
extension of the Standard Model of particle physics that incorporates an
additional scalar $\mathrm{SU}(2)_{\rm L}$ triplet coupled to the Higgs boson.
We show that within this scenario a strong electroweak FOPT is naturally
realised consistently with the measured $W$ boson mass-shift. Potentially
observable GW signatures imply the triplet mass scale to be TeV-ish,
consistently with the value preferred by the $W$ mass anomaly. This model can
be tested in future space-based interferometers such as LISA, DECIGO, BBO,
TianQin, TAIJI projects and in future colliders such as FCC, ILC, CEPC.
