  A simultaneous measurement of the three components of the top-quark and
top-antiquark polarisation vectors in $t$-channel single-top-quark production
is presented. This analysis is based on data from proton--proton collisions at
a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of
139 fb$^{-1}$, collected with the ATLAS detector at the LHC. Selected events
contain exactly one isolated electron or muon, large missing transverse
momentum and exactly two jets, one being $b$-tagged. Stringent selection
requirements are applied to discriminate $t$-channel single-top-quark events
from the background contributions. The top-quark and top-antiquark polarisation
vectors are measured from the distributions of the direction cosines of the
charged-lepton momentum in the top-quark rest frame. The three components of
the polarisation vector for the selected top-quark event sample are $P_{x'}
=0.01\pm0.18$, $P_{y'} = -0.029\pm0.027$, $P_{z'} =0.91\pm0.10$ and for the
top-antiquark event sample they are $P_{x'} = -0.02\pm0.20$, $P_{y'} =
-0.007\pm0.051$, $P_{z'} = -0.79\pm 0.16$. Normalised differential
cross-sections corrected to a fiducial region at the stable-particle level are
presented as a function of the charged-lepton angles for top-quark and
top-antiquark events inclusively and separately. These measurements are in
agreement with Standard Model predictions. The angular differential
cross-sections are used to derive bounds on the complex Wilson coefficient of
the dimension-six $\mathcal{O}_{tW}$ operator in the framework of an effective
field theory. The obtained bounds are $C_{tW} \in\ [-0.9,1.4]$ and $C_{itW} \in
[-0.8,0.2]$, both at 95\% confidence level.
