Results from a search for supersymmetry in events with four or more charged leptons (electrons, muons and taus) are presented. The analysis uses a data sample corresponding to 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton-proton collisions delivered by the Large Hadron Collider at $s=13$ TeV and recorded by the ATLAS detector. Four-lepton signal regions with up to two hadronically decaying taus are designed to target a range of supersymmetric scenarios that can be either enriched in or depleted of events involving the production and decay of a $Z$ boson. Data yields are consistent with Standard Model expectations and results are used to set upper limits on the event yields from processes beyond the Standard Model. Exclusion limits are set at the 95% confidence level in simplified models of General Gauge Mediated supersymmetry, where higgsino masses are excluded up to 295 GeV. In $R$-parity-violating simplified models with decays of the lightest supersymmetric particle to charged leptons, lower limits of 1.46 TeV, 1.06 TeV, and 2.25 TeV are placed on wino, slepton and gluino masses, respectively.