  The minimal free resolution of the coordinate ring of a complete intersection
in projective space is a Koszul complex on a regular sequence. In the product
of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which
sets of points have a virtual resolution that is a Koszul complex on a regular
sequence. This paper provides conditions on sets of points; some of which
guarantee the points have this property, and some of which guarantee the points
do not have this property.
