  We investigate the interplay of electronic correlations and spin-orbit
coupling (SOC) for a one-band and a two-band honeycomb lattice model. The main
difference between the two models concerning SOC is that in the one-band case
the SOC is a purely non-local term in the basis of the $p_z$ orbitals, whereas
in the two-band case with $p_x$ and $p_y$ as basis functions it is purely
local. In order to grasp the correlation effects on non-local spin-orbit
coupling, we apply the TRILEX approach that allows to calculate non-local
contributions to the self-energy approximatively. For the two-band case we
apply dynamical mean-field theory. In agreement with previous studies, we find
that for all parameter values in our study, the effect of correlations on the
spin-orbit coupling strength is that the bare effective SOC parameter is
increased. However, this increase is much weaker in the non-local than in the
local SOC case. Concerning the TRILEX method, we introduce the necessary
formulas for calculations with broken SU(2) symmetry.
