Three searches for the direct production of τ-sleptons or charginos and neutralinos in final states with at least two hadronically decaying τ-leptons are presented. For chargino and neutralino production, decays via intermediate τ-sleptons or W and h bosons are considered. The analysis uses a dataset of pp collisions corresponding to an integrated luminosity of 139 fb$^{−1}$, recorded with the ATLAS detector at the Large Hadron Collider at a centre-of-mass energy of 13 TeV. No significant deviation from the expected Standard Model background is observed and supersymmetric particle mass limits at 95% confidence level are obtained in simplified models. For direct production of $ {\overset{\sim }{\chi}}_1^{+}{\overset{\sim }{\chi}}_1^{-} $, chargino masses are excluded up to 970 GeV, while $ {\overset{\sim }{\chi}}_1^{\pm } $ and $ {\overset{\sim }{\chi}}_2^0 $ masses up to 1160 GeV (330 GeV) are excluded for $ {\overset{\sim }{\chi}}_1^{\pm }{\overset{\sim }{\chi}}_2^0/{\overset{\sim }{\chi}}_1^{+}{\overset{\sim }{\chi}}_1^{-} $ production with subsequent decays via τ-sleptons (W and h bosons). Masses of τ-sleptons up to 500 GeV are excluded for mass degenerate $ {\overset{\sim }{\tau}}_{L,R} $ scenarios and up to 425 GeV for $ {\overset{\sim }{\tau}}_L $-only scenarios. Sensitivity to $ {\overset{\sim }{\tau}}_R $-only scenarios from the ATLAS experiment is presented here for the first time, with $ {\overset{\sim }{\tau}}_R $ masses excluded up to 350 GeV.[graphic not available: see fulltext]