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A result by Bridson, Howie, Miller, and Short states that if $S$ is a
finitely presented subgroup of the direct product of free groups, then $S$ is
virtually a nilpotent extension of a direct product of free groups. Moreover,
if $S$ is a subgroup of type $FP_n$ of the direct product of $n$ free groups,
then the nilpotent extension is finite, so $S$ is actually virtually the direct
product of free groups.

In this paper, these results are generalized to $2$-dimensional coherent
right-angled Artin groups. More precisely, we show that a finitely presented
subgroup of the direct product of $2$-dimensional coherent RAAGs is still
virtually a nilpotent extension of a direct product of subgroups. If $S$ is
moreover a type $FP_n$ subgroup of the direct product of $n$ $2$-dimensional
coherent RAAGs, then $S$ is commensurable to a kernel of a character of a
direct product of subgroups.

Finally, we show that the multiple conjugacy problem and the membership
problem are decidable for finitely presented subgroups of direct products of
$2$-dimensional coherent RAAGs.