  We consider the hadroproduction of a massive charged lepton plus the
corresponding neutrino through the Drell-Yan mechanism. We present a new
computation of the mixed QCD-EW corrections to this process. The cancellation
of soft and collinear singularities is achieved by using a formulation of the
$q_T$ subtraction formalism derived from the next-to-next-to-leading order QCD
calculation for heavy-quark production. For the first time, all the real and
virtual contributions due to initial- and final-state radiation are
consistently included without any approximation, except for the finite part of
the two-loop virtual correction, which is computed in the pole approximation
and suitably improved through a reweighting procedure. We demonstrate that our
calculation is reliable in both on-shell and off-shell regions, thereby
providing the first prediction of the mixed QCD-EW corrections in the entire
region of the lepton transverse momentum. The computed corrections are in
qualitative agreement with what we obtain in a factorised approach of QCD and
EW corrections. At large values of the muon $p_T$, the mixed QCD-EW corrections
are negative and increase in size, to about $-20\%$ with respect to the
next-to-leading order QCD result at $p_T=500\,$GeV.
