  We consider the problem of determining if a pair of undirected graphs
$\langle G_\mathsf{V}, G_\mathsf{H} \rangle$, which share the same vertex set,
has a representation using opaque geometric shapes for vertices, and
vertical/horizontal visibility between shapes to determine the edges of
$G_\mathsf{V}$/$G_\mathsf{H}$. While such a simultaneous visibility
representation of two graphs can be determined efficiently if the direction of
the required visibility for each edge is provided (and the vertex shapes are
sufficiently simple), it was unclear if edge direction is critical for
efficiency. We show that the problem is $\mathsf{NP}$-complete without that
information, even for graphs that are only slightly more complex than paths. In
addition, we characterize which pairs of paths have simultaneous visibility
representations using fixed orientation L-shapes. This narrows the range of
possible graph families for which determining simultaneous visibility
representation is non-trivial yet not $\mathsf{NP}$-hard.
