The inclusive top-quark-pair production cross section 𝜎_𝑡𝑡̄ and its ratio to the 𝑍-boson production cross section have been measured in proton–proton collisions at √𝑠 = 13.6 TeV, using 29 fb⁻¹ of data collected in 2022 with the ATLAS experiment at the Large Hadron Collider. Using events with an opposite-charge electron-muon pair and 𝑏-tagged jets, and assuming Standard Model decays, the top-quark-pair production cross section is measured to be 𝜎_𝑡𝑡̄ = 850± 3(stat.) ± 18(syst.) ± 20(lumi.) pb. The ratio of the 𝑡𝑡̄ and the 𝑍-boson production cross sections is also measured, where the 𝑍-boson contribution is determined for inclusive 𝑒⁺𝑒⁻ and 𝜇⁺𝜇⁻ events in a fiducial phase space. The relative uncertainty on the ratio is reduced compared to the 𝑡𝑡̄ cross section, thanks to the cancellation of several systematic uncertainties. The result for the ratio, 𝑅_{𝑡𝑡̄∕𝑍} = 1.145 ± 0.003(stat.) ± 0.021(syst.)±0.002(lumi.) is consistent with the Standard Model prediction using the PDF4LHC21 PDF set.