Geometric classification of primes modulo a (bend) congruence

In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications of this result. (1) We prove an analogue of the strong Nullstellensatz for congruences with finite tropical basis. This extends the existing result of Jo\'o-Mincheva to cases, such as the bend congruence of a tropical(ized) ideal, where the congruence is not finitely generated. (2) We show that, if $I$ is the ideal of an affine variety not contained in the coordinate hyperplanes, then $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is cancellative. This result has applications to the integral closure (as per Tolliver) of $\mathbb{T}[x_1, \dots, x_n]/\operatorname{Bend}(\operatorname{trop} I)$ which we explore in a forthcoming paper. (3) We show that $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is the tropical function semiring on $\operatorname{trop} V(I)$, which creates a bridge between the algebraic approach to non-embedded tropicalization in the work of J. Song and the bend congruence approach of Giansiracusa-Giansiracusa and Maclagan-Rinc\'on. (4) As a consequence of one of our lemmas, we describe the closure of a polyhedron in a tropical toric variety even when the polyhedron is not compatible with the fan defining the tropical toric variety.