  The SYZ conjecture suggests a folklore that ``Lagrangian multi-sections are
mirror to holomorphic vector bundles". In this paper, we prove this folklore
for Lagrangian multi-sections inside the cotangent bundle of a vector space,
which are equivariantly mirror to complete toric varieties by the work of
Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem,
which asks whether one can construct an unobstructed Lagrangian multi-section
with asymptotic conditions prescribed by a tropical Lagrangian multi-section.
We solve the realization problem for tropical Lagrangian multi-sections over a
complete 2-dimensional fan that satisfy the so-called $N$-generic condition
with $N\geq 3$. As an application, we show that every rank 2 toric vector
bundle on the projective plane is mirror to a Lagrangian multi-section.

Symplectic Geometry