  We present the full next-to-leading-order electroweak and QCD corrections to
vector-boson scattering into a pair of off-shell opposite-sign W bosons
decaying into leptons of different flavour at the LHC. We include full
leading-order predictions for the irreducible background. Explicitly, we
investigate the process ${\rm pp \to e^+ \nu_e\mu^-\bar\nu_\mu jj} + X$ at
leading orders $O(\alpha^6)$, $O(\rm\alpha_s\alpha^5)$,
$O(\rm\alpha_s^2\alpha^4)$, supplemented by the loop-induced
$O(\rm\alpha_s^4\alpha^4)$ contribution, and at next-to-leading orders
$O(\alpha^7)$ and $O(\rm\alpha_s\alpha^6)$ in two setups providing fiducial
cross sections as well as differential distributions. We take full account of
photon-induced next-to-leading-order contributions, which prove to be non
negligible. With $-11.4\%$ and $-6.7\%$ in the two setups, the electroweak
corrections are smaller than for other vector-boson-scattering processes. This
can be traced back to the presence of the Higgs-boson resonance in the fiducial
phase space, whose effects we analyse within an additional unphysical, but
manifestly gauge-invariant setup. The QCD corrections amount to $-5.1\%$ and
$-21.6\%$ in the two setups. The large size of the latter correction, compared
to other vector-boson scattering processes, is explained by a very restrictive
definition of its fiducial phase space.
