A fully coupled implicit method has been developed for solving the viscous full multi-fluid equations, which incorporate transport and generation of mass and momentum for each component present in a system. This work presents stability analysis and representative computational results of this algorithm. The stability analyses demonstrate the performance of several iterative schemes applied to the solution of the linearized block system which arises in the fully implicit formulation. These include block Jacobi and symmetric block Gauss-Siedel schemes using two forms of relaxation. A hierarchy of increasing physical complexity is pursued, starting with one-dimensional, two-fluid systems with minimum inter-field dynamic coupling and no mass transfer. These analyses are then extended to systems employing physically important inter-field forces (drag, dispersion, virtual mass). The effects of mass transfer, multiple fields (i.e., more than 2), and multiple dimensions are considered. A two-fluid Navier-Stokes code has been developed, guided by the stability analyses. One-dimensional and two-dimensional results generated with this code are presented, which verify the validity of the stability analyses presented for the coupled scheme, and the effectiveness of the method for flows of engineering relevance.