  We construct examples of geometrically decomposable aspherical 4-manifolds
with non-zero signature. We show that all such 4-manifolds satisfy the
inequality (of Bogomolov--Miyaoka--Yau type) $\chi\geq 3|\sigma|$. We also
construct examples attaining the equality that are non-geometric and have
non-zero signature. Finally, we prove that for higher graph 4-manifolds, with
complex-hyperbolic vertices, the strict inequality always holds. Moreover, we
construct infinitely many examples of higher graph 4-manifolds with non-zero
signature and prove that the inequality is strict and sharp in this class.
