  The process ${\rm p}{\rm p}\to\mu^+\nu_\mu{\rm e}^+\nu_{\rm e}{\rm j}{\rm j}$
receives several contributions of different orders in the strong and
electroweak coupling constants. Using appropriate event selections, this
process is dominated by vector-boson scattering (VBS) and has recently been
measured at the LHC. It is thus of prime importance to estimate precisely each
contribution. In this article we compute for the first time the full NLO QCD
and electroweak corrections to VBS and its irreducible background processes
with realistic experimental cuts. We do not rely on approximations but use
complete amplitudes involving two different orders at tree level and three
different orders at one-loop level. Since we take into account all
interferences, at NLO level the corrections to the VBS process and to the
QCD-induced irreducible background process contribute at the same orders. Hence
the two processes cannot be unambiguously distinguished, and all contributions
to the $\mu^+\nu_\mu{\rm e}^+\nu_{\rm e}{\rm j}{\rm j}$ final state should be
preferably measured together.
