  We develop a combinatorial rule to compute the real geometry of type B
Schubert curves $S(\lambda_\bullet)$ in the orthogonal Grassmannian
$\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with
respect to orthogonal flags osculating the rational normal curve. Our results
are natural analogs of results previously known only in type A.
  First, using the type B Wronski map, we show that the real locus of the
Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy
operator $\omega$ defined as the commutator of jeu de taquin rectification and
promotion on skew shifted semistandard tableaux. We then introduce two
different algorithms to compute $\omega$ without rectifying the skew tableau.
The first uses recently-developed shifted tableau crystal operators, while the
second uses local switches much like jeu de taquin. The switching algorithm
further computes the K-theory coefficient of the Schubert curve: its
nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic
tableaux. The connection to K-theory also gives rise to a partial understanding
of the complex geometry of these curves.
