  The $k$-th exact-distance graph, of a graph $G$ has $V(G)$ as its vertex set,
and $xy$ as an edge if and only if the distance between $x$ and $y$ is
(exactly) $k$ in $G$. We consider two possible extensions of this notion for
signed graphs. Finding the chromatic number of a negative exact-distance square
of a signed graph is a weakening of the problem of finding the smallest target
graph to which the signed graph has a sign-preserving homomorphism. We study
the chromatic number of negative exact-distance graphs of signed graphs that
are planar, and also the relation of these chromatic numbers with the
generalised colouring numbers of the underlying graphs. Our results are related
to a theorem of Alon and Marshall about homomorphisms of signed graphs.
