A result of Baumslag and Roseblade states that a finitely presented subgroup
of the direct product of two free groups is virtually a direct product of free
groups. In this paper we generalise this result to the class of cyclic subgroup
separable graphs of groups with free abelian vertex groups and cyclic edge
groups. More precisely, we show that a finitely presented subgroup of the
direct product of two groups in this class is virtually H-by-(free abelian),
where H is the direct product of two groups in the class. In particular, our
result applies to 2-dimensional coherent right-angled Artin groups and
residually finite tubular groups. Furthermore, we show that the multiple
conjugacy problem and the membership problem are decidable for finitely
presented subgroups of the direct product of two 2-dimensional coherent
RAAGs