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A search for exotic decays of the Higgs boson into a pair of low-mass scalars that subsequently decay into 𝜏-leptons, 𝐻 → 𝑎𝑎 → 𝜏+ 𝜏−𝜏+ 𝜏−, is presented. In models with Yukawa-like couplings, the decay to 𝜏-leptons is favoured for light 𝑎-bosons, with mass in the range of 2𝑚𝜏 < 𝑚𝑎 < 2𝑚𝑏. Results are presented in the range of 4 GeV < 𝑚𝑎 < 15 GeV using the 140 f b−1 of proton–proton collisions at √𝑠 = 13 TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider. This search focuses on the scenario where, for both di-𝜏 pairs, one of the 𝜏-leptons decays to hadrons and neutrinos, while the other decays to a muon and neutrinos. In this mass range, the 𝑎 → 𝜏+ 𝜏− is Lorentz-boosted and a dedicated muon removal technique is used to reconstruct the di-𝜏 pairs. No significant excess above the Standard Model background prediction is observed. Upper limits on (𝜎(𝐻)∕𝜎SM (𝐻)) × Br(𝐻 → 𝑎𝑎 → 4𝜏) at 95 % confidence level are provided, ranging from 0.03 to 0.10 depending on the 𝑎-boson mass.