  Spin correlations in the production of top-antitop quark ($t\bar{t}$) pairs
at the Large Hadron Collider (LHC) are an experimentally verified prediction of
the Standard Model. In this paper, we compute the full spin density matrix for
$t\bar{t}Z$ production at next-to-leading order precision in QCD, for
center-of-mass energies of 13 and 14 TeV. We find that the additional emission
of a $Z$ boson leads to significantly different spin correlations with respect
to the $t\bar{t}$ case, and induces small longitudinal polarisations of the top
quarks. We further propose an analysis strategy that could lead to the
observation of spin correlations in $t\bar{t}Z$ events at the end of Run 3 of
the LHC, or possibly earlier by combining the ATLAS and CMS datasets. In
addition, we show that the pure angular information contained in the spin
density matrix provides novel constraints on the dimension-6 effective field
theory (EFT) operators relevant to the $t$-$Z$ interaction, without any
reference to the total production rates.
