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 doubly-charged $H^{\pm\pm}$ bosons with singly-charged $H^\pm$ bosons. No significant deviations from the Standard Model predictions are observed. $H^{\pm\pm}$ bosons are excluded at 95\% CL up to 350 GeV and 220 GeV for the pair and associated production modes, respectively.

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