  The topology of the Fermi surface of Sr2RuO4 is well described by
local-density approximation calculations with spin-orbit interaction, but the
relative size of its different sheets is not. By accounting for many-body
effects via dynamical mean-field theory, we show that the standard isotropic
Coulomb interaction alone worsens or does not correct this discrepancy. In
order to reproduce experiments, it is essential to account for the Coulomb
anisotropy. The latter is small but has strong effects; it competes with the
Coulomb-enhanced spin-orbit coupling and the isotropic Coulomb term in
determining the Fermi surface shape. Its effects are likely sizable in other
correlated multi-orbital systems. In addition, we find that the low-energy
self-energy matrix -- responsible for the reshaping of the Fermi surface --
sizably differ from the static Hartree-Fock limit. Finally, we find a strong
spin-orbital {entanglement}; this supports the view that the conventional
description of Cooper pairs via factorized spin and orbital part might not
apply to Sr2RuO4.
