  We present a calculation of the next-to-leading-order QCD and electroweak
corrections to WWW production with leptonically decaying $\mathrm{W}$ bosons at
the LHC, fully taking into account off-shell contributions, intermediary
resonances, and spin correlations. The contributions of the
quark-antiquark-induced electroweak correction to typical fiducial cross
sections at the LHC are of the order of 5-8% and grow to tens of percent in the
high-energy tails of distributions. We observe strong cancellations among the
positive quark-photon and negative quark-antiquark-induced electroweak
corrections. In addition to results based on full $2\to 6/7$-particle
next-to-leading-order matrix elements, we present a calculation based on the
triple-pole approximation, which expands the matrix elements around the poles
of three simultaneously resonant $\mathrm{W}$ bosons. The triple-pole
approximation performs particularly well for integrated cross sections and for
differential cross sections that are insensitive to off-shell effects, such as
angular and rapidity distributions.
