  Recent discrepancies between theoretical predictions and experimental data in
multi-lepton plus $b$-jets analyses for the $t\bar{t}W^\pm$ process, as
reported by the ATLAS collaboration, have indicated that more accurate
theoretical predictions and high precision observables are needed to constrain
numerous new physics scenarios in this channel. To this end we employ NLO QCD
computations with full off-shell top quark effects included to provide
theoretical predictions for the ${\cal R}=
\sigma_{t\bar{t}W^+}/\sigma_{t\bar{t}W^-}$ cross section ratio at the LHC with
$\sqrt{s}=13$ TeV. Depending on the transverse momentum cut on the $b$-jet we
obtain $2\% -3 \%$ theoretical precision on ${\cal R}$, which should help to
shed some light on new physics effects that can reveal themselves only once
sufficiently precise Standard Model theoretical predictions are available.
Furthermore, triggered by these discrepancies we reexamine the charge asymmetry
of the top quark and its decay products in the $t\bar{t}W^\pm$ production
process. In the case of charge asymmetries, that are uniquely sensitive to the
chiral nature of possible new physics in this channel, theoretical
uncertainties below $15\%$ are obtained. Additionally, the impact of the top
quark decay modelling is scrutinised by explicit comparison with predictions in
the narrow-width approximation.
