  In this article, we study minimal isometric immersions of K\"ahler manifolds
into product of two real space forms. We analyse the obstruction conditions to
the existence of pluriharmonic isometric immersions of a K\"ahler manifold into
those spaces and we prove that the only ones into
$\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{H}^{m-1}\times \mathbb{R}$ are
the minimal isometric immersions of Riemannian surfaces. Futhermore, we show
that the existence of a minimal isometric immersion of a K\"ahler manifold
$M^{2n}$ into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{S}^{m-k}\times
\mathbb{H}^k$ imposes strong restrictions on the Ricci and scalar curvatures of
$M^{2n}$. In this direction, we characterise some cases as either isometric
immersions with parallel second fundamental form or anti-pluriharmonic
isometric immersions.
