Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces

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