The Hermite method for real root counting generalizes to the multivariate case if your system of polynomial inequalities has only a finite number of COMPLEX roots. this was shown by Becker and Wörmann and independently by Pedersen, see:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.9539

The Hermite method for real root counting generalizes to the multivariate case if your system of polynomial inequalities has only a finite number of COMPLEX roots. See for example the article "Radical computations of zero-dimensional ideals and real root counting" by Becker and Wörmann and the references therein.