  A degeneration of a smooth projective curve to a strongly stable curve gives
rise to a specialization map from divisors on curves to divisors on graphs. In
this paper we show that this specialization behaves well under the presence of
real structures. In particular we study real linear systems on graphs with a
real structure and we prove results on them comparable to results in the
classical theory of real curves. We also consider generalizations to metric
graphs and tropical curves.
