  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 $\text{p}\text{p}\to
{\text{e}}^+\nu_{\text{e}}\,\mu^-\bar{\nu}_\mu\,\tau^+\nu_\tau\,{\text{b}}\,\bar{\text{b}}$,
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.
