  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.
