  We present a new Monte Carlo event generator for the production of a
top-quark pair in association with a $W^\pm$ boson at hadron colliders in the
POWHEG-BOX framework. We consider the next-to-leading-order QCD corrections to
the $pp\to t\bar{t} W^\pm$ cross section, corresponding to the
$\mathcal{O}(\alpha_s^3\alpha)$ and $\mathcal{O}(\alpha_s\alpha^3)$ terms in
the perturbative expansion of the parton-level cross section, and model the
decays of $W$ and top quarks at leading order retaining spin correlations. The
fixed-order QCD calculation is further interfaced with the Pythia8
parton-shower event generator via the POWHEG method as implemented in the
POWHEG-BOX. The corresponding code is now part of the public repository of the
POWHEG-BOX. We perform a comparison of different event generators for both the
case of inclusive production and the case of the two same-sign leptons
signature at the Large Hadron Collider operating at a center-of-mass energy of
$13$ TeV. We investigate theoretical uncertainties in the modelling of the
fiducial volume stemming from missing higher-order corrections, the different
parton shower matching schemes, and the modelling of decays. We find that the
subleading contribution at $\mathcal{O}(\alpha_s\alpha^3)$ is particularly
sensitive to differences in the matching scheme and higher-order parton shower
effects. We observe that in particular jet observables can differ quite visibly
although these differences play only a subordinate role in the description of
physical observables once all contributions are combined.
