Using the methods of numerical algebraic geometry, one can compute a numerical irreducible decomposition of the solution set of polynomial systems. This decomposition describes the enitre solution set and its breakup into irreducible pieces over complex Euclidean space. However, in engineering or science, it is common that only the real solutions are of interest. A single complex component may contain multiple real components, some possibly having lower dimension in the reals than the dimension of the complex component that contains them. We present an algorithm for finding all real solutions inside the pure-one-dimensional complex solution set of a polynomial system. The algorithm finds a numerical approximation to all isolated real solutions and a description of all real curves in a Morse-like representation consisting of vertices with edges connecting them. The work presented in this talk has been done in collaboration with Ye Lu, Daniel Bates, and Andrew Sommese.