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 W+ boson in proton-proton collisions. As the calculation is based on the full matrix elements for the process mathrm{pp}to {mathrm{e}}^{+}{v}_{mathrm{e}}{mu}^{-}{overline{v}}_{mu }{tau}^{+}{v}_{tau}mathrm{b}overline{mathrm{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.