  We study the hadroproduction of a $Wb$ pair in association with a light jet,
focusing on the dominant $t$-channel contribution and including exactly at the
matrix-element level all non-resonant and off-shell effects induced by the
finite top-quark width. Our simulations are accurate to the next-to-leading
order in QCD, and are matched to the HERWIG6 and PYTHIA8 parton showers through
the MC@NLO method. We present phenomenological results relevant to the 8 TeV
LHC, and carry out a thorough comparison to the case of on-shell $t$-channel
single-top production. We formulate our approach so that it can be applied to
the general case of matrix elements that feature coloured intermediate
resonances and are matched to parton showers.
