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We investigate full off-shell effects in $t\bar{t}b\bar{b}$ production in the dilepton channel at the LHC with the center-of-mass energy $\sqrt{s} = 13$ TeV. Specifically, we compute NLO QCD corrections to the $pp \to e^+ ν_e μ^- \barν_μb \bar{b} b \bar{b} + X$ process and provide a prescription for $b$-jet identification to distinguish prompt $b$ jets from $b$ jets originating from the decay of the top quarks. As an important irreducible background to $pp \to t\bar{t}H (H\to b\bar{b})$, $t\bar{t}$ production in association with two prompt $b$ jets is a primary source of uncertainty in the measurement of $t\bar{t}H (H\to b\bar{b})$. In quantifying full off-shell effects, we perform comparisons between the state-of-the-art full off-shell computation and the calculation in the narrow width approximation. The former includes all double-, single- and non-resonant Feynman diagrams, interferences as well as finite-width effects of the top quarks and $W$ gauge bosons. The latter restricts the unstable top quarks and $W$ gauge bosons to on-shell states and includes for the first time NLO QCD corrections to both production and decays. We observe that full off-shell effects are subdominant compared to the scale uncertainties for the integrated fiducial cross section and for the majority of differential observables in the phase-space regions that we investigated. However, for a number of observables related to beyond the Standard Model searches, full off-shell effects are significant. Furthermore, with our $b$-jet labelling prescription, the prompt $b$ jets and the $b$ jets from top-quark decays can be successfully disentangled.