  In a statistical physics context, inverse problems consist in determining
microscopic interactions such that a system reaches a predefined collective
state. A complex collective state may be prescribed by specifying the overlap
distribution between microscopic configurations, a notion originally introduced
in the context of disordered systems like spin-glasses. We show that in spite
of the absence of disorder, nonequilibrium spin models exhibiting spontaneous
magnetization oscillations provide a benchmark to prescribe a non-trivial
overlap distribution with continuous support, qualitatively analogous to the
ones found in disordered systems with full replica symmetry breaking. The
overlap distribution can be explicitly tailored to take a broad range of
predefined shapes by monitoring the spin dynamics. The presence of a
non-trivial overlap distribution is traced back to an average over infinitely
many pure states, a feature shared with spin-glasses, although the structure of
pure states is here much simpler.
