  We study surface subgroups of $\mathrm{SL}(4,\mathbb R)$ acting convex
cocompactly on $\mathbb R \textrm P^3$ with image in the coaffine group. The
boundary of the convex core is stratified, and the one dimensional strata form
a pair of bending laminations. We show that the bending data on each component
consist of a convex $\mathbb R \textrm P^2$ structure and an affine measured
lamination depending on the underlying convex projective structure on $S$ with
(Hitchin) holonomy $\rho: \pi_1S \to \mathrm{SL}(3,\mathbb R)$. We study the
space $\mathcal {ML}^\rho(S)$ of bending data compatible with $\rho$ and prove
that its projectivization is a sphere of dimension $6g-7$.
