The first measurement of the CP structure of the Yukawa coupling between the Higgs boson and $\tau$ leptons is presented. The measurement is based on data collected in proton-proton collisions at \sqrt{s}=13~\mathrm{TeV} by the CMS experiment at the CERN LHC in 2016, 2017 and 2018, corresponding to an integrated luminosity of 137~\mathrm{fb}^{-1}. Events are selected where one $\tau$ decays to a muon and the other hadronically, and where both $\tau$ leptons decay hadronically. These are the most sensitive decay modes for this analysis and together cover about $50\%$ of all Higgs-to-tau decays. The analysis uses the angular correlation between the decay planes of $\tau$ leptons produced in Higgs boson decays. Machine learning techniques are deployed to distinguish between signal and background events, and dedicated analysis techniques are used to optimise the reconstruction of the $\tau$ decay planes. The mixing angle between CP-even and CP-odd $\tau$ Yukawa couplings was found to be $4\pm 17^{\circ}$, compared to an expected uncertainty of $\pm 23^{\circ}$ at the $68\%$ confidence level, while at the $95\%$ confidence level the observed (expected) uncertainties were $\pm36^{\circ}$ $(\pm 55)^{\circ}$. The observed (expected) significance of the separation between the CP-even and CP-odd hypotheses is $3.2$ $(2.3)$ standard deviations. The results are compatible with predictions for the standard model Higgs boson.

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