I will describe a family S of Schubert problems on the Grassmannian, defined using flags osculating (tangent to) the rational normal curve at a chosen set of marked points.
This family is very well-behaved (for example, it is Cohen-Macaulay), particularly when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) showed that the solutions are then "as real as possible", and Speyer (2014) extended the construction to stable curves, showing that the real locus of S is a smooth cover of the moduli space of real stable curves. Moreover, the monodromy of the cover has a remarkable description in terms of Young tableaux and Schützenberger's jeu de taquin.
I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.