  We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar
graphs each with maximum degree 4 and the same vertex set, do they admit an
OrthoSEFE, that is, is there an assignment of the vertices to grid points and
of the edges to paths on the grid such that the same edges in distinct graphs
are assigned the same path and such that the assignment induces a planar
orthogonal drawing of each of the $k$ graphs?
  We show that the problem is NP-complete for $k \geq 3$ even if the shared
graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$
even if the shared graph consists of a cycle and of isolated vertices. Whereas
the problem is polynomial-time solvable for $k=2$ when the union graph has
maximum degree five and the shared graph is biconnected. Further, when the
shared graph is biconnected and has sunflower intersection, we show that every
positive instance has an OrthoSEFE with at most three bends per edge.
