  We study Higgs boson production in association with a top quark and a $W$
boson at the LHC. At NLO in QCD, $tWH$ interferes with $t\bar t H$ and a
procedure to meaningfully separate the two processes needs to be employed. In
order to define $tWH$ production for both total rates and differential
distributions, we consider the diagram removal and diagram subtraction
techniques that have been previously proposed for treating intermediate
resonances at NLO, in particular in the context of $tW$ production. These
techniques feature approximations that need to be carefully taken into account
when theoretical predictions are compared to experimental measurements. To this
aim, we first critically revisit the $tW$ process, for which an extensive
literature exists and where an analogous interference with $t \bar t$
production takes place. We then provide robust results for total and
differential cross sections for $tW$ and $tWH$ at 13 TeV, also matching
short-distance events to a parton shower. We formulate a reliable prescription
to estimate the theoretical uncertainties, including those associated to the
very definition of the process at NLO. Finally, we study the sensitivity to a
non-Standard-Model relative phase between the Higgs couplings to the top quark
and to the $W$ boson in $tWH$ production.
