Navier-Stokes equations in the whole space R^3 subject to an anisotropic viscosity and a random perturbation of multiplicative type is described. By adding a term of Brinkman-Forchheimer type to the model, existence and uniqueness of global weak solutions in the PDE sense are proved. These are strong solutions in the probability sense. The convective term given in terms of the Brinkman-Forchheirmer provides some extra regularity in the space L^{2α+2} (R^3), with α > 1. As a consequence, the nonlinear term has better properties which allows to prove uniqueness. The proof of existence is performed through a control method. A Large Deviations Principle is given and proven at the end of the paper.