  In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the
variety $\mathbb{X}_{k} := {\rm Gr}(k,n) \times \mathbb{C}^\times$ which
tropicalizes to the affine crystal structure on rectangular tableaux with $n-k$
rows. In this sequel, we define and study the geometric $R$-matrix, a
birational map $R : \mathbb{X}_{k_1} \times \mathbb{X}_{k_2} \rightarrow
\mathbb{X}_{k_2} \times \mathbb{X}_{k_1}$ which tropicalizes to the
combinatorial $R$-matrix on pairs of rectangular tableaux. We show that $R$ is
an isomorphism of geometric crystals, and that it satisfies the Yang--Baxter
relation. In the case where both tableaux have one row, we recover a birational
action of the symmetric group that has appeared in the literature in a number
of contexts. We also define a rational function $E : \mathbb{X}_{k_1} \times
\mathbb{X}_{k_2} \rightarrow \mathbb{C}$ which tropicalizes to the coenergy
function from affine crystal theory.
  Most of the properties of the geometric $R$-matrix follow from the fact that
it gives the unique solution to a certain equation of matrices in the loop
group ${\rm GL}_n(\mathbb{C}(\lambda))$.
