$\newcommand\Q{\mathbb Q} \newcommand\R{\mathbb R} \newcommand\P{\mathbb P}$Let $K$ be a Galois field extension of $\Q$. Define $L:=K\cap \R$ and suppose that there are three distinct points $P_1,P_2,P_3 \in \P^2(L)$ that are not collinear. Is it true that the set of points $$X:=\bigcup_{i=1}^3\{\sigma(P_i):\sigma\in\mathrm{Gal}(K/\Q) \} $$ are in general position in $\P^2(K)$? For what conditions of $P_1,P_2,P_3\in \P^2(L)$ can we conclude that the above generated set consists of points in general position in $\P^2(K)$? I think I can reduce the question to the affine plane as well.
Actually I don't even know this for just quadratic field extensions $K$ (for a moment I could care less about the field not being real). My aim is to determine the minimum degree in the vanishing ideal of $X$.