Schubert curves are the spaces of solutions to certain one-dimensional Schubert problems involving flags osculating the rational normal curve. The real locus of a Schubert curve is known to be a natural covering space of $RP^1$, so its real geometry is fully characterized by the monodromy of the cover. It is also possible, using K-theoretic Schubert calculus, to relate the real locus to the overall (complex) Riemann surface. We present a local algorithm for computing the monodromy operator in terms of Jeu de Taquin-like operations on certain skew Young tableaux, and use it to provide purely combinatorial proofs of some of the connections to K-theory. We will also explore partial progress in this direction in the Type C setting of the orthogonal grassmannian. This is joint work with Jake Levinson.