On one hand the real locus of a complex elliptic curve is the intersection of a plane with a torus (i.e. a torus embedded in $\mathbb{C}^2$ plus infinity). And an elliptic curve has no cusps or self-intersections. But the intersection of a plane with a non-singular torus can have those.

I suppose those intersections of a complex cubic curve with a plane which can (up to some complex projective change of coordinates) give a real locus cannot have cusps or self intersections unless the cubic does too.

But is that right? Is it easy to see, once you see it? If it is not right what is?