  We present next-to-leading order (NLO) calculations including QCD and
electroweak (EW) corrections for $2\ell2\nu$ diboson signatures with two
opposite-charge leptons and two neutrinos. Specifically, we study the processes
$pp\to e^+ \mu^- \nu_{e}\bar\nu_{\mu}$ and $pp\to e^+ e^-\nu\bar\nu$, including
all relevant off-shell diboson channels, $W^+W^-$, $ZZ$, $\gamma Z$, as well as
non-resonant contributions. Photon-induced processes are computed at NLO EW,
and we discuss subtle differences related to the definition and the
renormalisation of the coupling $\alpha$ for processes with initial- and
final-state photons. All calculations are performed within the automated
Munich/Sherpa+OpenLoops frameworks, and we provide numerical predictions for
the LHC at 13 TeV. The behaviour of the corrections is investigated with
emphasis on the high-energy regime, where NLO EW effects can amount to tens of
percent due to large Sudakov logarithms. The interplay between $WW$ and $ZZ$
contributions to the same-flavour channel, $pp\to e^+ e^-\nu\bar\nu$, is
discussed in detail, and a quantitative analysis of photon-induced
contributions is presented. Finally, we consider approximations that account
for all sources of large logarithms, at high and low energy, by combining
virtual EW corrections with a YFS soft-photon resummation or a QED parton
shower.
