On the Fano Surface of a Real Cubic $M$-Threefold

We prove that the Fano surface which parameterizes the set of straight lines on a real $M$-threefold of degree 3 is also an $M$-surface. The converse statement is also true. The proof is based on the results about the intermediate Jacobian of a complex cubic threefold due to C. Clemens and P. Griffiths.