  A new definition of prime congruences in additively idempotent semirings is
given using twisted products. This class turns out to exhibit some analogous
properties to the prime ideals of commutative rings. In order to establish a
good notion of radical congruences it is shown that the intersection of all
primes of a semiring can be characterized by certain twisted power formulas. A
complete description of prime congruences is given in the polynomial and
Laurent polynomial semirings over the tropical semifield ${\pmb T}$, the
semifield $\mathbb{Z}_{max}$ and the two element semifield $\mathbb{B}$. The
minimal primes of these semirings correspond to monomial orderings, and their
intersection is the congruence that identifies polynomials that have the same
Newton polytope. It is then shown that every finitely generated congruence in
each of these cases is an intersection of prime congruences with quotients of
Krull dimension $1$. An improvement of a result of A. Bertram and R. Easton
from 2013 is proven which can be regarded as a Nullstellensatz for tropical
polynomials.
