Take the tour ×  In Alain Roberts "Elliptic curves: notes from
postgraduate lectures given in Lausanne 1971/72" page 11
(available on google books unless you already tried to read
another chapter), there is a hand drawn picture of a real
2-dimensional torus and a real plane, which topologically
represent the way a complex cubic (with two real components) and
the real projective plane sit in the complex projective plane.
Taking the picture on face value, one should be able to project
an open subset of the complex projective plane to
$\mathbb{R}^3$, so that there is some real line $L$ that passes
through the "doughnut" defined by the image of the complex
cubic.  I tried to reproduce this picture on a computer, using
the map $\mathbb{CP}^2\to\mathbb{R}^7$ given by   projecting to
various $\mathbb{R}^3$s, and looking for $L$ by trial and error;
all in vain. Which brings me to....     • Is there such a line
(the map I used does not send the real projective plane to a
plane, so it does not have to be the case even