  We provide the complete set of predictions needed to achieve NLO accuracy in
the Standard Model Effective Field Theory at dimension six for Higgs production
in gluon fusion. In particular, we compute for the first time the contribution
of the chromomagnetic operator $ \bar Q_L \Phi \sigma q_R G$ at NLO in QCD,
which entails two-loop virtual and one-loop real contributions, as well as
renormalisation and mixing with the Yukawa operator $\Phi^\dagger \Phi\, \bar
Q_L \Phi q_R$ and the gluon-fusion operator $\Phi^\dagger \Phi\, GG$. Focusing
on the top-quark-Higgs couplings, we consider the phenomenological impact of
the NLO corrections in constraining the three relevant operators by
implementing the results into the MadGraph5_aMC@NLO framework. This allows us
to compute total cross sections as well as to perform event generation at NLO
that can be directly employed in experimental analyses.
