I'm trying to solve a big system of:

quadratic equations with coefficients in $\mathbb{Z}$, each in 6 variables, quadratic inequalities with coefficients in $\mathbb{Z}$, each in 6 variables, and linear inequalities of the form $x_i>0$.

There are on the order of a couple of dozen equations in my system. I know that the system has exactly one solution in the real numbers, and the solution is algebraic. Some of the equations might be extraneous.

I don't know anything about the solution space over $\mathbb{C}$ or over the entire field of algebraic numbers. Presumably there are more solutions there, though I don't know this for sure, either.

I need an exact solution, so for every variable, I want either a closed expression for it, or an irreducible polynomial in one variable that has it as a root.

I've looked into MAGMA for this, hoping to solve the system using a computer algebra implementation of Groebner bases. However, I am having a couple of problems.

My questions are:

How might I include the inequalities in my system? Is there a way to make them into equalities? I don't mind introducing extra variables.

Is there a computer algebra system that will find Groebner bases over the real algebraic numbers? Or, is this something I could possibly implement with a little work from existing solvers that work over the algebraic closure of $\mathbb{Q}$?