  This paper presents measurements of top-antitop quark pair ($t\bar{t}$)
production in association with additional $b$-jets. The analysis utilises 140
fb$^{-1}$ of proton-proton collision data collected with the ATLAS detector at
the Large Hadron Collider at a centre-of-mass energy of 13 TeV. Fiducial
cross-sections are extracted in a final state featuring one electron and one
muon, with at least three or four $b$-jets. Results are presented at the
particle level for both integrated cross-sections and normalised differential
cross-sections, as functions of global event properties, jet kinematics, and
$b$-jet pair properties. Observable quantities characterising $b$-jets
originating from the top quark decay and additional $b$-jets are also measured
at the particle level, after correcting for detector effects. The measured
integrated fiducial cross-sections are consistent with $t\bar{t}b\bar{b}$
predictions from various next-to-leading-order matrix element calculations
matched to a parton shower within the uncertainties of the predictions.
State-of-the-art theoretical predictions are compared with the differential
measurements; none of them simultaneously describes all observables.
Differences between any two predictions are smaller than the measurement
uncertainties for most observables.
