A search for exotic decays of the 125 GeV Higgs boson into a pair of new spin-0 particles, H → aa, where one decays into a photon pair and the other into a τ-lepton pair, is presented. Hadronic decays of the τ-leptons are considered and reconstructed using a dedicated tagger for collimated τ-lepton pairs. The search uses 140 fb⁻¹ of proton-proton collision data at a centre-of-mass energy of s = 13 TeV recorded between 2015 and 2018 by the ATLAS experiment at the Large Hadron Collider. The search is performed in the mass range of the a boson between 10 GeV and 60 GeV. No significant excess of events is observed above the Standard Model background expectation. Model-independent upper limits at 95% confidence level are set on the branching ratio of the Higgs boson to the γγττ final state, B(H → aa → γγττ), ranging from 0.2% to 2%, depending on the a-boson mass hypothesis.