In the paper we study the existence of balanced metrics of Hodge–Riemann type on non-Kähler complex manifolds. We first find some general obstructions, for instance that a Hodge–Riemann balanced manifold of complex dimension n has to be \((n - 2)\) -Kähler. Then, we focus on the case of compact quotients of Lie groups by lattices, endowed with an invariant complex structure. In particular, we prove non-existence results on non-Kähler complex parallelizable manifolds and some classes of solvmanifolds, and we show that the only nilmanifolds admitting invariant structures of this type are tori. Finally, we construct the first non-Kähler example of a Hodge–Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by \(\mathbb {C}\) .
