This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from the simplex domain to the product-of-simplex domain, which is directly applicable to solving Nash equilibria. The fixed-point bundle for VIs on the product-of-simplex domain reveals a composite fiber bundle structure. The key innovation is to construct an equivalent VI on the simplex domain and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, the predictor-corrector path-following algorithm for the VI on the product-of-simplex domain is shown to inherit the convergence guarantee of the simplex-domain framework, namely, global convergence with linear gap reduction near solutions. Numerical experiments on 5600 randomly generated instances with dimensions ranging from 2-player 128-action to 128-player 2-action demonstrate robust performance. The algorithm converges in every tested instance.