  Electroweak (EW) triboson production processes with at least one heavy gauge
boson are of increasing interest at the Large Hadron Collider (LHC) as direct
precision probes of one of the least-tested sectors of the Standard Model (SM),
the quartic couplings of the EW gauge bosons. These processes therefore offer
promising opportunities for searches for indirect signals of Beyond-the-SM
(BSM) physics. In this paper, we present results for fiducial cross sections at
next-to-leading-order (NLO) EW and NLO QCD to $p\;p\to
e^{+}\;\nu_{e}\;\mu^{+}\;\mu^{-}\;\gamma$ at the 13 TeV LHC. This signature
includes the triboson production process $p\;p \to W^{+}\;Z\;\gamma$ with
leptonic decays, $W^{+} \to e^{+}\;\nu_{e}$ and $Z \to \mu^{+}\;\mu^{-}$. The
computation is based on the complete set of leading-order (LO) contributions of
$\mathcal{O}(\alpha^5)$ and on NLO EW and NLO QCD cross sections of
$\mathcal{O}(\alpha^6)$ and $\mathcal{O}(\alpha^5 \alpha_\text{s})$,
respectively, and thus off-shell effects, spin correlations and non-resonance
contributions are fully taken into account. We construct a Monte Carlo
framework which provides total and differential cross sections for a chosen set
of basic analysis cuts. We find that while NLO EW corrections enhance the
fiducial LO total cross section by only $1\%$, they can significantly change
some distributions in certain kinematic regions. For example, the relative NLO
EW corrections to the muon transverse momentum distribution at 500 GeV amounts
to $-20\%$. To illustrate how missing NLO EW corrections could masquerade as
BSM physics, we show examples for the impact of dimension-8 operators in the SM
Effective Field Theory framework on selected kinematic distributions.
