I need to prove the existence of real solution(s) to a system of 7 polynomial equalities in 14 variables up to degree 6 for a range of values of 5 real parameters $t_1,t_2,t_3,x_0,x_3$, with $t_1>0, \quad t_2\geq 0, \quad t_3>0,\quad x_3\geq x_0$. I've included the (messy) system at the bottom of this post to give an example of its structure. Empirically I expect solutions to always exist, as the problem can be reformulated as an SDP for some values of $t_1, t_2, t_3, x_0, x_3$ and numerically solved for (so far) any random combination of these parameters within this set.
I'm a little lost as to how best to approach this problem, can anybody suggest an approach that may help me? I see the Real Nullstellensatz can be used to prove the existence of solutions in all 19 variables, but how can I reformulate this to show there exist solutions for all values of these five parameters?
I've also tried computing the Groebner basis for this system in various orders, but after hours of running this does not finish, and I suspect won't in my lifetime.
$$\left(3 \left(q_{a11}^2+q_{b11}^2\right)+4 \left(-q_{a11}^2-q_{b11}^2+s_{a12}^2+s_{b12}^2+s_{c12}^2\right)\right) t_1^3+(6 s_{a11} s_{a12}+6 s_{b11} s_{b12}+6 s_{c11} s_{c12}) t_1^2+\left(2 s_{a11}^2+2 s_{b11}^2+2 s_{c11}^2\right) t_1=0\\\left(s_{a12}^2+s_{b12}^2+s_{c12}^2\right) t_1^4+(2 s_{a11} s_{a12}+2 s_{b11} s_{b12}+2 s_{c11} s_{c12}) t_1^3+\left(s_{a11}^2+s_{b11}^2+s_{c11}^2\right) t_1^2=0\\\left(-\frac{q_{a11}^2}{5}+\frac{s_{a12}^2}{5}+\frac{s_{b12}^2}{5}+\frac{s_{c12}^2}{5}+\frac{1}{4} \left(q_{a11}^2+q_{b11}^2\right)-\frac{q_{b11}^2}{5}\right) t_1^5-\frac{1}{60} \left(3 q_{a11}^2+3 q_{b11}^2+12 s_{a12}^2+12 s_{b12}^2+12 s_{c12}^2\right) t_1^5+\left(\frac{s_{a11} s_{a12}}{2}+\frac{s_{b11} s_{b12}}{2}+\frac{s_{c11} s_{c12}}{2}\right) t_1^4-\frac{1}{60} (30 s_{a11} s_{a12}+30 s_{b11} s_{b12}+30 s_{c11} s_{c12}) t_1^4+\left(\frac{s_{a11}^2}{3}+\frac{s_{b11}^2}{3}+\frac{s_{c11}^2}{3}\right) t_1^3-\frac{1}{60} \left(20 s_{a11}^2+20 s_{b11}^2+20 s_{c11}^2\right) t_1^3=0\\\left(-\frac{q_{a11}^2}{30}+\frac{s_{a12}^2}{30}+\frac{s_{b12}^2}{30}+\frac{s_{c12}^2}{30}+\frac{1}{20} \left(q_{a11}^2+q_{b11}^2\right)-\frac{q_{b11}^2}{30}\right) t_1^6+\left(\frac{s_{a11} s_{a12}}{10}+\frac{s_{b11} s_{b12}}{10}+\frac{s_{c11} s_{c12}}{10}\right) t_1^5-\left(-\frac{t_2 q_{a11}^2}{20}+\left(-\frac{q_{a11}^2}{20}-\frac{s_{a12}^2}{5}-\frac{s_{b12}^2}{5}-\frac{s_{c12}^2}{5}-\frac{q_{b11}^2}{20}\right) t_3-\frac{s_{a12}^2 t_2}{5}-\frac{s_{b12}^2 t_2}{5}-\frac{s_{c12}^2 t_2}{5}-\frac{q_{b11}^2 t_2}{20}\right) t_1^5+\left(\frac{s_{a11}^2}{12}+\frac{s_{b11}^2}{12}+\frac{s_{c11}^2}{12}\right) t_1^4-\left(-\frac{1}{2} s_{a11} s_{a12} t_2-\frac{s_{b11} s_{b12} t_2}{2}-\frac{s_{c11} s_{c12} t_2}{2}+\left(-\frac{s_{a11} s_{a12}}{2}-\frac{s_{b11} s_{b12}}{2}-\frac{s_{c11} s_{c12}}{2}\right) t_3\right) t_1^4-\left(-\frac{t_2 s_{a11}^2}{3}+\left(-\frac{s_{a11}^2}{3}-\frac{s_{b11}^2}{3}-\frac{s_{c11}^2}{3}\right) t_3-\frac{s_{b11}^2 t_2}{3}-\frac{s_{c11}^2 t_2}{3}\right) t_1^3-\left(\frac{s_{a32}^2}{30}+\frac{s_{b32}^2}{30}+\frac{s_{c32}^2}{30}\right) t_3^6-\left(\frac{s_{a31} s_{a32}}{10}+\frac{s_{b31} s_{b32}}{10}+\frac{s_{c31} s_{c32}}{10}\right) t_3^5-\left(\frac{s_{a31}^2}{12}+\frac{s_{b31}^2}{12}+\frac{s_{c31}^2}{12}\right) t_3^4+x0-x3=0\\4 \left(-s_{a32}^2-s_{b32}^2-s_{c32}^2\right) t_3^3+(-6 s_{a31} s_{a32}-6 s_{b31} s_{b32}-6 s_{c31} s_{c32}) t_3^2+\left(-2 s_{a31}^2-2 s_{b31}^2-2 s_{c31}^2\right) t_3=0\\\left(-s_{a32}^2-s_{b32}^2-s_{c32}^2\right) t_3^4+(-2 s_{a31} s_{a32}-2 s_{b31} s_{b32}-2 s_{c31} s_{c32}) t_3^3+\left(-s_{a31}^2-s_{b31}^2-s_{c31}^2\right) t_3^2=0\\\left(\frac{q_{a11}^2}{20}+\frac{q_{b11}^2}{20}+\frac{s_{a12}^2}{5}+\frac{s_{b12}^2}{5}+\frac{s_{c12}^2}{5}\right) t_1^5+\left(\frac{s_{a11} s_{a12}}{2}+\frac{s_{b11} s_{b12}}{2}+\frac{s_{c11} s_{c12}}{2}\right) t_1^4+\left(\frac{s_{a11}^2}{3}+\frac{s_{b11}^2}{3}+\frac{s_{c11}^2}{3}\right) t_1^3+\left(-\frac{s_{a32}^2}{5}-\frac{s_{b32}^2}{5}-\frac{s_{c32}^2}{5}\right) t_3^5+\left(-\frac{s_{a31} s_{a32}}{2}-\frac{s_{b31} s_{b32}}{2}-\frac{s_{c31} s_{c32}}{2}\right) t_3^4+\left(-\frac{s_{a31}^2}{3}-\frac{s_{b31}^2}{3}-\frac{s_{c31}^2}{3}\right) t_3^3=0$$