  The Bethe algebras for the Gaudin model act on the multiplicity space of
tensor products of irreducible $ \mathfrak{gl}_r $-modules and have simple
spectrum over real points. This fact is proved by Mukhin, Tarasov and Varchenko
who also develop a relationship to Schubert intersections over real points. We
use an extension to $ \overline{M}_{0,n+1}(\mathbb{R}) $ of these Schubert
intersections, constructed by Speyer, to calculate the monodromy of the
spectrum of the Bethe algebras. We show this monodromy is described by the
action of the cactus group $ J_n $ on tensor products of irreducible $
\mathfrak{gl}_r $-crystals.
