  We present a new continuation algorithm to find all nondegenerate real
solutions to a system of polynomial equations. Unlike homotopy methods, it is
not based on a deformation of the system; instead, it traces real curves
connecting the solutions of one system of equations to those of another,
eventually leading to the desired real solutions. It also differs from homotopy
methods in that it follows only real paths and computes no complex solutions of
the original equations. The number of curves traced is bounded by the fewnomial
bound for real solutions, and the method takes advantage of any slack in that
bound.
