  Building off of work of Harada, Nowroozi, and Van Tuyl which provided
particular length two virtual resolutions for finite sets of points in
$\mathbb{P}^1\times\mathbb{P}^1$, we prove that the vast majority of virtual
resolutions of a pair for minimal elements of the multigraded regularity in
this setting are of Hilbert-Burch type. We give explicit descriptions of these
short virtual resolutions that depend only on the number of points. Moreover,
despite initial evidence, we show that these virtual resolutions are not always
short, and we give sufficient conditions for when they are length three.
