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 χ~1+χ~1−, chargino masses are excluded up to 970 GeV, while χ~1± and χ~20 masses up to 1160 GeV (330 GeV) are excluded for χ~1±χ~20/χ~1+χ~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 τ~L, R scenarios and up to 425 GeV for τ~L-only scenarios. Sensitivity to τ~R-only scenarios from the ATLAS experiment is presented here for the first time, with τ~R masses excluded up to 350 GeV.