  We use ideas from the Strominger--Yau--Zaslow conjecture and microlocal sheaf
theory to define graded modules associated with permissible
$C^{\infty}$-divisors on compact tropical manifolds. A $C^{\infty}$-divisor is
a generalization of a Lagrangian section on an integral affine manifold. The
group of $C^{\infty}$-divisors on a tropical manifold surjects onto the Picard
group. We also prove a Riemann--Roch formula for compact tropical curves and
integral affine manifolds admitting Hessian forms. Our approach differs from
the tropical Riemann--Roch theorem established by Gathmann and Kerber.
