  The inclusive top-quark-pair production cross section $\sigma_{t\bar{t}}$ and
its ratio to the $Z$-boson production cross section have been measured in
proton--proton collisions at $\sqrt{s} = 13.6$ 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 $\sigma_{t\bar{t}} = 850 \pm 3\mathrm{(stat.)}\pm
18\mathrm{(syst.)}\pm 20\mathrm{(lumi.)}$ pb. The ratio of the $t\bar{t}$ and
the $Z$-boson production cross sections is also measured, where the $Z$-boson
contribution is determined for inclusive $e^+e^-$ and $\mu^+\mu^-$ events in a
fiducial phase space. The relative uncertainty on the ratio is reduced compared
to the $t\bar{t}$ cross section, thanks to the cancellation of several
systematic uncertainties. The result for the ratio, $R_{t\bar{t}/Z} = 1.145 \pm
0.003\mathrm{(stat.)}\pm 0.021\mathrm{(syst.)}\pm 0.002\mathrm{(lumi.)}$ is
consistent with the Standard Model prediction using the PDF4LHC21 PDF set.
