  We investigate the virtual specialness of a compact cube complex $X$ that
splits as a graph of nonpositively curved cube complexes. We prove virtual
specialness of $X$ when each vertex space of $X$ has word-hyperbolic $\pi_1$
and $\pi_1X$ has ``finite stature'' relative to its edge groups. The results
generalize the motivating case when tree $\times$ tree lattices are virtual
products.
