In  we have considered a family of almost anti-Hermitian structures (G,J) on the tangent bundle TM of a Riemannian manifold (M,g), where the almost complex structure J is a natural lift of g to TM interchanging the vertical and horizontal distributions VTM and HTM and the metric G is a natural lift of g of Sasaki type, with the property of being anti-Hermitian with respect to J. Next, we have studied the conditions under which (TM,G,J) belongs to one of the eight classes of anti-Hermitian structures obtained in the classification in . In this paper, we study some geometric properties of the anti-Kählerian structure obtained in . In fact we prove that it is Einstein. This result offers nice examples of anti-Kählerian Einstein manifolds studied in .