Say I have a system of multivariate polynomials, $[P_{1}, ... ,P_{n}]$. I'm interested in determining whether it has a real-valued solution, but don't necessarily want to find any solutions, or enumerate solutions. I expect that often I'll be dealing with systems that have infinitely many real solutions.
In an algebraically closed field, this can be answered by taking a reduced Groebner basis of $\langle P_{1}, ... P_{n} \rangle$. If I'm not mistaken, the system has no solutions iff its reduced Groebner basis is {1}.
Is there an analogous algorithm for determining whether there are real solutions to a system? If not, what algorithms are available?