A search for dark matter produced in association with a Higgs boson in final states with two hadronically decaying $\tau$-leptons and missing transverse momentum is presented. The analysis uses $139$ fb$^{-1}$ of proton-proton collision data at $\sqrt{s}=13$ TeV collected by the ATLAS experiment at the Large Hadron Collider between 2015 and 2018. No evidence for physics beyond the Standard Model is found. The results are interpreted in terms of a 2HDM+$a$ model. Exclusion limits at 95% confidence level are derived. Model-independent limits are also set on the visible cross section for processes beyond the Standard Model producing missing transverse momentum in association with a Higgs boson decaying to $\tau$-leptons.
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