The real Plücker–Klein map

We consider the generalized Plücker–Klein map from the set of all real marked biquadrics to the set of real Kummer varieties. We find a necessary and sufficient condition on a real marked biquadric in order that the corresponding real Kimmer variety be isomorphic to the real Kummer variety induced by the real Jacobian of a double covering of the pencil of quadrics through the given biquadric. We also give a deformation classification of the real Plücker–Klein map.