  A search for charged Higgs bosons decaying into $W^\pm W^\pm$ or $W^\pm Z$
bosons is performed, involving experimental signatures with two leptons of the
same charge, or three or four leptons with a variety of charge combinations,
missing transverse momentum and jets. A data sample of proton-proton collisions
at a centre-of-mass energy of 13 TeV recorded with the ATLAS detector at the
Large Hadron Collider between 2015 and 2018 is used. The data correspond to a
total integrated luminosity of 139 fb$^{-1}$. The search is guided by a type-II
seesaw model that extends the scalar sector of the Standard Model with a scalar
triplet, leading to a phenomenology that includes doubly and singly charged
Higgs bosons. Two scenarios are explored, corresponding to the pair production
of doubly charged $H^{\pm\pm}$ bosons, or the associated production of a doubly
charged $H^{\pm\pm}$ boson and a singly charged $H^\pm$ boson. No significant
deviations from the Standard Model predictions are observed. $H^{\pm\pm}$
bosons are excluded at 95% confidence level up to 350 GeV and 230 GeV for the
pair and associated production modes, respectively.
