Would you please recommend a computer program that could give quick answer Yes/No for the question: does there exist a real solution of a given system of polynomial equations with integer coefficients. I will need it to solve a huge list of such systems, each of them is over $\mathbb{R}^6$, has $6$ equations of degree $4$ or less. Coefficients are also quite small. Maple's Triangularize procedure for most of the cases works too long, so applying it for big list is almost impossible.

Have you thought obtain a Groebner basis of the set of defining polynomials of your system? Maybe, it could simplify considerably the aspect of the system, and you would need a little time to calculate the real solutions of the system. For that, you could use the free software wxMaxima. If you don't have it, you can download in http://maxima.sourceforge.net/download.html. I would like post this comment just as a comment and not as an answer, but I don't know how to do it. 

