ag.algebraic geometry - Real solutions to underdetermined system of polynomial equations

If you're dealing with finding solutions to a specific system, the traditional method is cylindrical algebraic decomposition, described for instance in Algorithms in Real Algebraic Geometry
by Saugata Basu, Richard Pollack, Marie-Françoise Roy downloadable HERE. This method is widely implemented in many computer algebra packages. An alternative method is the critical point method, which was implemented in Maple by Mohab Safey El Din (LINK). (Both methods eventually use Groebner bases b the way.)

This might let you experiment, but I doubt that these methods alone can give you all you need to know about a family of systems. Solving over the reals is harder than over the complexes (because the interplay between algebra and geometry is less straightforward). There is active research on the algorithmic aspects of the real Nullstellensatz, (e.g. M. Coste, H. Lombardi, M.-F. Roy. Dynamical method in algebra: efective Nullstellensatze, Annals of Pure and Applied Logic, 111, 203-256 (2001)), but the complexity is even worse than Groebner bases.