  For any subgroup of $\mathrm{SL}(3,\mathbb{R})\ltimes\mathbb{R}^3$ obtained
by adding a translation part to a subgroup of $\mathrm{SL}(3,\mathbb{R})$ which
is the fundamental group of a finite-volume convex projective surface, we first
show that under a natural condition on the translation parts of parabolic
elements, the affine action of the group on $\mathbb{R}^3$ has convex domains
of discontinuity that are regular in a certain sense, generalizing a result of
Mess for globally hyperbolic flat spacetimes. We then classify all these
domains and show that the quotient of each of them is an affine manifold
foliated by convex surfaces with constant affine Gaussian curvature. The proof
is based on a correspondence between the geometry of an affine space endowed
with a convex cone and the geometry of a convex tube domain. As an independent
result, we show that the moduli space of such groups is a vector bundle over
the moduli space of finite-volume convex projective structures, with rank
equaling the dimension of the Teichm\"uller space.
