The inclusive top-quark-pair production cross section σtt¯ 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 σtt¯=850±3(stat.)±18(syst.)±20(lumi.) pb. The ratio of the tt¯ 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 tt¯ cross section, thanks to the cancellation of several systematic uncertainties. The result for the ratio, Rtt¯/Z=1.145±0.003(stat.)±0.021(syst.)±0.002(lumi.) is consistent with the Standard Model prediction using the PDF4LHC21 PDF set.