  In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real
Kahler submanifold of codimension 2, if with complex dimension > 2, would be
either holomorphic, or a cylinder, or complex ruled. In this article, we
generalize their result to real analytic complete real Kahler submanifolds of
codimension 4. The conclusion is that such the submanifold, if with complex
dimension > 4, would be either partially holomorphic, or a cylinder, or a
twisted cylinder in the sense that the complex relative nullity foliation is
contained in a strictly larger holomorphic foliation, whose leaves are
cylinders. We also examine the question of when such a submanifold is complex
ruled.
