  We report on the calculation of the next-to-leading order QCD corrections to
the production of a $t\bar{t}$ pair in association with two heavy-flavour jets.
We concentrate on the di-lepton $t\bar{t}$ decay channel at the LHC with
$\sqrt{s}=13$ TeV. The computation is based on $pp \to e^+ \nu_e\,
\mu^-\bar{\nu}_\mu\, b\bar{b} \,b\bar{b}$ matrix elements and includes all
resonant and non-resonant diagrams, interferences and off-shell effects of the
top quark and the $W$ gauge boson. As it is customary for such studies, results
are presented in the form of inclusive and differential fiducial cross
sections. We extensively investigate the dependence of our results upon
variation of renormalisation and factorisation scales and parton distribution
functions in the quest for an accurate estimate of the theoretical
uncertainties. We additionally study the impact of the contributions induced by
the bottom-quark parton density. Results presented here are particularly
relevant for measurements of $t\bar{t}H(H\to b\bar{b})$ and the determination
of the Higgs coupling to the top quark. In addition, they might be used for
precise measurements of the top-quark fiducial cross sections and to
investigate top-quark decay modelling at the LHC.
