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. ) electronic configuration and Ru atoms at sites with D 4h symmetry; due to the layered structure the Ru t 2g xz and yz bands are almost one-dimensional and very narrow, with a band width W xz = W yz about half as large as that of the two-dimensional Ru xy band, W xy . Experimentally, the Fermi surface of Sr 2 RuO 4 has been studied via both the de Haas-van Alphen technique [14-16] and angle-resolved photoemission spectroscopy (ARPES) [17][18][19][20]. It is made (Fig. 1) by three sheets, the electron-like γ (xy band) and β (xz, yz bands) sheets and the hole-like α sheet (xz, yz bands). Theoretically, ab-initio calculations based on the local-density approximation (LDA) qualitatively reproduce the FS topology, provided that the spin-orbit (SO) interaction is taken into account [22,23]. Indeed, several experiments point to a sizable SO coupling [1,25,26]. These calculations fail, however, in describing the relative size of the sheets, suggesting that perhaps many-body effects play a key role. The relevance of the Coulomb interaction for the electronic properties of Sr 2 RuO 4 , as well as its interplay with bands of different width, was shown early on via model many-body studies [21]. More recently, LDA+DMFT (local-density approximation + dy- namical mean-field theory) calculations have emphasized the interplay of Coulomb interaction and t 2g crystal field (CF) [9,27], and the role of the Hund's rule coupling [10]. LDA+slave-boson calculations point to SO effects on the correlated bands [28]. It remains however unclear to what extent many-body effects actually modify the Fermi surface, and how they compete with other effects. In this Letter, by using the LDA+DMFT method with SO interaction, we investigate, for the first time, the interplay between Coulomb repulsion, spin-orbit and sym-