I have been thinking about the exercise 11.3.C of Vakil's FOAG. The answers posted here Trouble with Vakil's FOAG exercise 11.3.C are not clarifying enough for me, since I have though of the following example: Take $\mathbb{P^2_{\mathbb{R}}}$ with coordinates $(x:y:z)$ and (with the notation of the exercise) set
$H = V(z),\quad X = V((x^2+y^2-z^2))$
Then, these 2 clearly don't meet in $\mathbb{P^2_{\mathbb{R}}}$ (although they meet in $\mathbb{P^2_{\mathbb{C}}}$).
What am I missing here? Does anyone have a proof for this?
Thank you very much!