In an externally applied magnetic field, ultrapure crystals of the bilayer compound Sr3 Ru2 O7 undergo a metamagnetic transition below a critical temperature, T, which varies as a function of the angle between the magnetic field H and the Ru-O planes. Moreover, T approaches zero when H is perpendicular to the planes. This putative "metamagnetic quantum critical point," however, is pre-empted by a nematic fluid phase with order one resistive anisotropy in the ab plane. In a "realistic" bilayer model with moderate strength local Coulomb interactions, the existence of a sharp divergence of the electronic density of states near a van Hove singularity of the quasi-one-dimensional bands, and the presence of spin-orbit coupling results in a mean-field phase diagram which accounts for many of these experimentally observed phenomena. Although the spin-orbit coupling is not overly strong, it destroys the otherwise near-perfect Fermi-surface nesting and hence suppresses spin-density-wave ordering. © 2009 The American Physical Society.