  We introduce the notion of tropical Lagrangian multi-sections over a
$2$-dimensional integral affine manifold $B$ with singularities, and use them
to study the reconstruction problem for higher rank locally free sheaves over
Calabi-Yau surfaces. To certain tropical Lagrangian multi-sections $\mathbb{L}$
over $B$, which are explicitly constructed by prescribing local models around
the ramification points, we construct locally free sheaves
$\mathcal{E}_0(\mathbb{L},{\bf{k}}_s)$ over the singular projective scheme
$X_0(B,\mathscr{P},s)$ associated to $B$ equipped with a polyhedral
decomposition $\mathscr{P}$ and a gluing data $s$. We then find combinatorial
conditions on such an $\mathbb{L}$ under which the sheaf
$\mathcal{E}_0(\mathbb{L},{\bf{k}}_s)$ is simple. This produces explicit
examples of smoothable pairs
$(X_0(B,\mathscr{P},s),\mathcal{E}_0(\mathbb{L},{\bf{k}}_s))$ in dimension 2.
