  We present a new calculation of $W^+ W^-$ production in the semi-exclusive
channel, that is either with intact outgoing protons or rapidity gaps present
in the final state, and with no colour flow between the colliding protons. This
study provides the first complete prediction of the $W^+ W^-$ semi-exclusive
cross section, as well as the breakdown between elastic and proton dissociative
channels. It combines the structure function calculation for a precise
modelling of the region of low momentum transfers with a parton-level
calculation in the region of high momentum transfers. The survival factor
probability of no additional proton-proton interactions is fully accounted for,
including its kinematic and process dependence. We analyse in detail the role
that the pure photon-initiated ($\gamma\gamma \to W^+ W^-$) subprocess plays, a
comparison that is only viable by working in the electroweak axial gauge. In
this way, we find that the dominance of this is not complete in the proton
dissociative cases, although once $Z$-initiated production is included a
significantly better matching to the complete calculation is achieved. A direct
consequence of this is that the relative elastic, single and double
dissociative fractions are in general different in comparison to the case of
lepton pair production. We present a direct comparison to the recent ATLAS data
on semi-exclusive $W^+ W^-$ production, finding excellent agreement within
uncertainties. Our calculation is provided in the publicly available SuperChic
4.1 Monte Carlo (MC) generator, and can be passed to a general purpose MC for
showering and hadronization of the final state.
