  This letter details and discusses the next-to-leading order QCD corrections
to $t$-channel electro-weak $W^+ b j$ production, where finite top-width
effects are consistently taken into account. The computation is done within the
aMC@NLO framework and includes both resonant and non-resonant contributions as
well as interferences between the two. Results are presented for the LHC and
compared to those of the narrow-width approximation and effective theory
approaches.
