Based on the joint work with Matteo Gallet The diffeomorphism type of small hyperplane arrangement is combinatorially determined, I would like to characterize those line arrangements with up to 10 lines that have combinatorially-determined topology of their complement manifold.

In particular, I need some help in order to to better understand the conjugate action on the complex realization space of a matroid.

$\textbf{Set up.}$ Let $f_{1},\ldots,f_{m}\in\mathbb{R}[t_{1},\ldots,t_{d}]$ be polynomials with real coefficients and let $X$ be the complex variety defined by $X=\{P\in\mathbb{C}^{d}\mid f_{i}(P)=0, 1\leq i\leq m\}$. Let $\sim$ be the equivalence relation defined on $X$ by $P\sim Q\Longleftrightarrow P=Q\text{ or }P=\overline{Q}$ (here $\overline{Q}$ is the point obtained conjugating componentwise the entries of $Q$). Finally, let us consider the quotient $X/\sim$.

$\textbf{Question.}$ Is that possible to explicit describe the quotient $X/\sim$, perhaps in terms of a certain family of equations?