  Recently a slight tension between data and predictions has been reported in
$t\bar{t}W$ production by both the CMS and ATLAS collaborations. We revisit the
theoretical predictions for this process, focussing on the following two
effects. We disentangle various effects that lead to asymmetries among the
leptonic decay products of the (anti-)top quarks and $W$ bosons, for which we
find that the spin correlations in the top-quark pair are the dominant source.
We also discuss the impact of the large, formally subleading, electroweak
corrections to $t\bar{t}W$ production at the LHC. We find that this effect
changes the $t\bar{t}W$ cross section significantly in the signature
phase-space regions, and should therefore be included differentially in the
theory to data comparisons.
