  KKL observer design consists in finding a smooth change of coordinates
transforming the system dynamics into a linear filter of the output. The state
of the original system is then reconstructed by implementing this filter from
any initial condition and left-inverting the transformation, under a
\textit{backward-distinguishability} property. In this paper, we consider the
case where the latter assumption does not hold, namely when distinct solutions
may generate the same output, and thus be indistinguishable. The KKL
transformation is no longer injective and its ``left-inverse'' is thus allowed
to be set-valued, yielding a set-valued KKL observer. Assuming the
transformation is full-rank and its preimage has constant cardinality, we show
the existence of a globally defined set-valued left-inverse that is Lipschitz
in the Hausdorff sense. Leveraging on recent results linking this left-inverse
with the \textit{backward-indistinguishable sets}, we show that the set-valued
KKL observer converges in the Hausdorff sense to the backward-indistinguishable
set of the system solution. When, additionally, a given output is generated by
a specific number of solutions not converging to each other, we show that the
designed observer asymptotically reconstructs each of those solutions. Finally,
the different assumptions are discussed and illustrated via examples.
