  We present next-to-leading order QCD and electroweak (EW) theory predictions
for $V+2\,$jet production, with $V=Z,W^{\pm}$, considering both the QCD and EW
production modes and their interference. We focus on phase-space regions where
$V+2\,$jet production is dominated by vector-boson fusion, and where these
processes yield the dominant irreducible backgrounds in searches for invisible
Higgs boson decays. Predictions at parton level are provided together with
detailed prescriptions for their implementation in experimental analyses based
on the reweighting of Monte Carlo samples. The key idea is that, exploiting
accurate data for $W+2\,$jet production in combination with a theory-driven
extrapolation to the $Z+2\,$jet process can lead to a determination of the
irreducible background at the few-percent level. Particular attention is
devoted to the estimate of the residual theoretical uncertainties due to
unknown higher-order QCD and EW effects and their correlation between the
different $V+2\,$jet processes, which is key to improve the sensitivity to
invisible Higgs decays.
