A search for the production of a Higgs boson in association with a single top quark
in multi-lepton final states including $\tau$-leptons is presented, using an integrated
luminosity of 140 fb$^{-1}$ of proton--proton collision data at a centre-of-mass energy
of $\sqrt{s} = 13$ TeV collected by the ATLAS detector at the LHC.
The analysis targets the $H \to \tau\tau$ decay, where at least one $\tau$-lepton decays
hadronically, to indicate the presence of the Higgs boson.
No significant excess over the expected background is observed, and an observed\,(expected)
upper limit at the 95\% confidence level on the signal strength is set at 8.6\,(9.3) times
the Standard Model prediction.

This measurement is combined with previous publications of the ATLAS Collaboration targeting
the same production process but in different final states, using 140 fb$^{-1}$ of
proton--proton collision data at $\sqrt{s} = 13$ TeV and 164 fb$^{-1}$ at 13.6 TeV.
The combined measurement of the signal strength yields
$\mu_{tH} = 3.3^{\,+1.7}_{\,-1.5} = 3.3 \pm 1.2\,\text{(stat.)}\,^{+1.1}_{-1.0}\,\text{(sys.)}$
times the Standard Model expectation, corresponding to an observed (expected) $tH$
significance of $2.3\,(0.8)$ standard deviations.
The combined observed\,(expected) upper limit at the 95% confidence level on the signal
strength is set at $6.1\,(2.7)$ times the Standard Model prediction.
This combination represents the most precise measurement of $tH$ production to date.