Constraining the $CP$ structure of Higgs-fermion couplings with a global LHC fit, the electron EDM and baryogenesis

$CP$ violation in the Higgs couplings to fermions is an intriguing, but not yet extensively explored possibility. We use inclusive and differential LHC Higgs boson measurements to fit the $CP$ structure of the Higgs Yukawa couplings. Starting with simple effective models featuring $CP$ violation in a single Higgs-fermion coupling, we probe well-motivated models with up to nine free parameters. We also investigate the complementarity of LHC constraints with the electron electric dipole moment bound, taking into account the possibility of a modified electron Yukawa coupling, and assess to which extent $CP$ violation in the Higgs-fermion couplings can contribute to the observed baryon asymmetry of the universe. Even after including the recent analysis of angular correlations in H → τ${}^{+}$τ${}^{-}$ decays, we find that a complex tau Yukawa coupling alone may be able to account for the observed baryon asymmetry, but with large uncertainties in the baryogenesis calculation. A combination of complex top and bottom quark Yukawa couplings yields a result four times larger than the sum of their separate contributions, but remains insufficient to account for the observed baryon asymmetry.