Numerical tropical line bundles and toric b-divisors

We study the relationship between line bundles on tropical compactifications of a very affine variety $Y$ and toric b-divisors on the associated tropical variety ${\rm Trop}(Y)$. By focusing on numerical equivalence classes, we construct a natural injective map from the group of numerical tropical line bundles on $Y$ to the space of toric b-divisors modulo linear equivalence. Moreover, we show that this map restricts to a bijection between the tropical nef cone of $Y$ and the set of toric b-divisors that are b-Cartier and tropically nef. This provides a higher-dimensional generalization of Baker's specialization for curves and clarifies the birational nature of tropical line bundles. We also discuss the kernel of the map from line bundles to numerical tropical line bundles, which encodes the continuous moduli lost in tropicalization.