  Let X be a plane in a torus over an algebraically closed field K, with
tropicalization the matroidal fan Sigma. In this paper we present an algorithm
which completely solves the question whether a given one-dimensional balanced
polyhedral complex in Sigma is relatively realizable, i.e. whether it is the
tropicalization of an algebraic curve, over the field of Puiseux series over K,
in X. The algorithm implies that the space of all such relatively realizable
curves of fixed degree is an abstract polyhedral set.
  In the case when X is a general plane in 3-space, we use the idea of this
algorithm to prove some necessary and some sufficient conditions for relative
realizability. For 1-dimensional polyhedral complexes in Sigma that have
exactly one bounded edge, passing through the origin, these necessary and
sufficient conditions coincide, so that they give a complete non-algorithmic
solution of the relative realizability problem.
