  We consider Schubert problems with respect to flags osculating the rational
normal curve. These problems are of special interest when the osculation points
are all real -- in this case, for zero-dimensional Schubert problems, the
solutions are "as real as possible". Recent work by Speyer has extended the
theory to the moduli space $\overline{M_{0,r}}$, allowing the points to
collide. These give rise to smooth covers of $\overline{M_{0,r}}(\mathbb{R})$,
with structure and monodromy described by Young tableaux and jeu de taquin.
  In this paper, we give analogous results on one-dimensional Schubert problems
over $\overline{M_{0,r}}$. Their (real) geometry turns out to be described by
orbits of Sch\"{u}tzenberger promotion and a related operation involving
tableau evacuation. Over $M_{0,r}$, our results show that the real points of
the solution curves are smooth.
  We also find a new identity involving `first-order' K-theoretic
Littlewood-Richardson coefficients, for which there does not appear to be a
known combinatorial proof.
