  In a previous paper we developed the notions of th-independence and \th-ranks
which define a geometric independence relation in a class of theories which we
called ``rosy''. We proved that rosy theories include simple and o-minimal
theories and that for any theory for which the stable forking conjecture was
true, \th-forking coincides with forking independence.
  In this article, we continue to study properties of th-forking and find more
examples of rosy theories. Among the new properties we prove in this paper are
some alternative characterizations of rosy theories and some tools to prove and
analyze rosiness in particular cases. Finally, we use this to find two examples
of rosy non simple theories: pseudo real closed fields (PRC-fields) and the
uniform companion of a large differential field defined by Marcus Tressl.
