Real varieties associated to complex varieties

For any irreducible and reduced compact complex space or any integral projective variety, let Xσ be its complex conjugate. Let S(X) be the set of all W such that W×Wσ X×Xσ. We say that S(X) is trivial if for every W S(X) there are Y,Z such that X Y×Z and W Y×Zσ. A torsion bundle over a torus is the total space of a fibre bundle with finite structure group over a positive dimensional torus. Here we prove that if X has no torsion bundle over a torus as a factor, then S(X) is trivial.