We present results of a computation of NLO QCD corrections to the production of an off-shell top--antitop pair in association with an off-shell $text{W}^+$ boson in proton--proton collisions. As the calculation is based on the full matrix elements for the process $text{p}text{p}to {text{e}}^+ u_{text{e}},mu^-bar{ u}_mu,tau^+ u_tau,{text{b}},bar{text{b}}$, all off-shell, spin-correlation, and interference effects are included. The NLO QCD corrections are about $20%$ for the integrated cross-section. Using a dynamical scale, the corrections to most distributions are at the same level, while some distributions show much larger $K$-factors in suppressed regions of phase space. We have performed a second calculation based on a double-pole approximation. While the corresponding results agree with the full calculation within few per cent for integrated cross-sections, the discrepancy can reach $10%$ and more in regions of phase space that are not dominated by top--antitop production. As a consequence, on-shell calculations should only be trusted to this level of accuracy.
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