Can I systematically solve in $\mathbb{R}^4$ the following system without using Grobner basis algorithm ? If not, can I find the exact number of solutions ?
$$ \begin{equation*} \left\{ \begin{aligned} X_1^2+X_2^2=a\\ X_1X_3+X_2X_4=b\\ X_3^2+X_4^2+eX_1+fX_2=c\\ eX_3+fX_4=d \end{aligned} \right. \end{equation*} $$
$a,b,c,d,e,f$ are known reals constants
One can use resultants. Given two equations of degrees $k_1, k_2$,
$$F_1(x_1,x_2,\dots,x_m) = 0$$
$$F_2(x_1,x_2,\dots,x_n) = 0$$
the resultant can eliminate one variable between them. By applying it repeatedly to a system, you can reduce it to just one equation. (It is computationally expensive, but if the degrees $k_i$ of your equations are small enough, then it is feasible.)
For your system, I find one can resolve it into a sextic in $x_4$. (I used Mathematica). Thus, in general, you have six solutions. However, the transformation,
$$x_4 = \frac{ez+df}{e^2+f^2}$$
simplifies it to the form,
$$z^6+c_1z^4+c_2z^2+c_3=0$$
so the sextic has a solvable Galois group and your $x_i$ in fact can be expressed in radicals.