  Precision measurements of top-associated Higgs production are an important
ingredient to unravel the $\mathcal{CP}$ nature of the Higgs boson. In this
work, we constraint the $\mathcal{CP}$ nature of the top-Yukawa coupling taking
into account all relevant inclusive and differential Higgs boson measurements.
Based upon this fit, we show that it is crucial to disentangle single- and
di-top-associated Higgs production for tightening indirect constraints on a
$\mathcal{CP}$-odd top-Yukawa coupling in the future. In this context, we
propose an analysis strategy for measuring $tH$ production at the HL-LHC
without relying on assumptions about the Higgs $\mathcal{CP}$ character.
