The Fano variety of the Clebsch diagonal cubic surface (x^3+y^3+z^3+w^3)-(x+y+z+w)^3 consists of 27 lines, all defined over the real field. Here is a picture of an affine portion of the surface with the 27 lines drawn.

You can see a plaster model of this constructed in the 1880's at the mathematical model collection of the University of Turin. In particular at the bottom of that page you can investigate details about these 27 lines.