  We present the first complete calculation of mixed QCD-electroweak
corrections to the production of on-shell $Z$ bosons in hadron collisions and
their decays to massless charged leptons. Our computation is fully differential
with respect to final state QCD partons and resolved photons, allowing us to
compute any infra-red safe observable pertinent to the $pp \to Z \to l^+ l^-$
process in the approximation that the $Z$ boson is on shell. Although mixed
QCD-electroweak corrections are small, at about the per mill level, we observe
that the interplay between QCD-QED and QCD-weak contributions is subtle and
observable-dependent. It is therefore not possible to avoid computing one or
the other if ${\cal O}(\alpha_{EW} \alpha_s)$ precision is desired.
