  Coxeter groups possess an associative operation, called variously the
Demazure, greedy, or $0$-Hecke product. For symmetric groups, this product has
an amusing formulation as matrix multiplication in the min-plus (tropical)
semiring of two matrices associated to the permutations. We prove that this
min-plus formulation extends to furnish a Demazure product on a much larger
group of integer permutations, consisting of all permutations that change the
sign of finite many integers. We prove several alternative descriptions of this
product and some useful properties of it. These results were developed in
service of future applications to Brill-Noether theory of algebraic and
tropical curves; the connection is surveyed in an appendix.
