  In this article we discuss a minimal extension of the Inert Doublet Model
(IDM) with an effective $CP$-violating $D=6$ operator, involving the inert
Higgs and weak gauge bosons, that can lift it to a fully realistic setup for
creating the baryon asymmetry of the Universe (BAU). Avoiding the need to stick
to an explicit completion, we investigate the potential of such an operator to
give rise to the measured BAU during a multi-step electroweak phase transition
(EWPhT) while sustaining a viable DM candidate in agreement with the measured
relic abundance. We find that the explored extension of the IDM can account
quantitatively for both DM and for baryogenesis and has quite unique virtues,
as we will argue. It can thus serve as a benchmark for a minimal realistic
extension of the SM that solves some of its shortcomings and could represent
the low energy limit of a larger set of viable completions. After discussing
the impact of a further class of operators that open the possibility for a
larger mass splitting (enhancing the EWPhT) while generating the full relic
abundance also for heavy inert-Higgs DM, we ultimately provide a quantitative
evaluation of the induced lepton electric dipole moments in the minimal
benchmark for the BAU. These arise here at the two-loop level and are therefore
less problematic compared to the ones that emerge when inducing $CP$ violation
via an operator involving the SM-like Higgs.
