  The top quark as the heaviest particle in the Standard Model (SM) defines an
important mass scale for Higgs physics and the electroweak scale itself. It is
therefore a well-motivated degree of freedom which could reveal the presence of
new interactions beyond the SM. Correlating modifications of the top-Higgs
interactions in the 2-Higgs-Doublet Model (2HDM), we analyse effective field
theory deformations of these interactions from the point of view of a strong
first-order electroweak phase transition (SFOEWPT). We show that such
modifications are compatible with current Higgs data and that an SFOEWPT can be
tantamount to a current overestimate of exotic Higgs searches' sensitivity at
the LHC in $t\bar t$ and four top quark final states. We argue that these
searches remain robust from the point of accidental signal-background
interference so that the current experimental strategy might well lead to
2HDM-like discoveries in the near future.
