The occurrence and significance of complex characteristics in two-phase flow equation systems are clarified by a detailed analysis of separated two-phase flow between two parallel plates. The basic system of one-dimensional two-phase flow equations for this problem possesses complex characteristics, exhibits unbounded instabilities in the short-wavelength limit, and constitutes an improperly posed initial value problem. These difficulties have led some workers to propose major modifications to the basic equation system. We show that the relatively minor modification of introducing surface tension is sufficient to render the characteristics real, stabilize short-wavelength disturbances, and produce a properly posed problem. For a given value of the surface tension, the basic equation system thus modified is shown to correctly predict the evolution of small-amplitude disturbances having wavelengths long compared to the plate spacing. A formula is given for the artificial surface tension necessary to stabilize wavelengths on the order of the mesh spacing in a finite difference numerical calculation. A brief discussion is given concerning the expected behavior of surface tension as compared to viscosity in the nonlinear regime. The general relation between characteristics and stability is discussed in an appendix.