  Inclusive hadronic decays of the $\tau$ lepton are very interesting from the
phenomenological point of view since they give access to the CKM matrix
elements $V_{ud}$ and $V_{us}$. In this paper, for the first time, by employing
the HLT method for hadronic smeared spectral densities we compute on the
lattice the inclusive decay rate of the processes $\tau \to X_{ud}\, \nu_\tau$,
where $X_{ud}$ is a generic hadronic state with $\bar{u} d$ flavor quantum
numbers. Our computation, which avoids any recourse to OPE and/or perturbative
approximations, is carried out in isospin symmetric $N_{f}=2+1+1$ lattice QCD
at physical quark masses, using ensembles produced by the ETMC at three lattice
spacings and two volumes. All uncertainties, except for isospin breaking
effects, are taken into account and a result with a subpercent error is
obtained for $|V_{ud}|$, which is nicely consistent with the current world
average. These findings validate our approach and also motivate the inclusion
of isospin breaking corrections and its extension to the inclusive decay $\tau
\to X_{us}\, \nu_\tau$, paving the way towards a high-precision first
principles determination of $|V_{us}|$ and $|V_{ud}|$ from inclusive $\tau$
decay.
