Abstract
We study the martingale problem for two mutually competing superprocesses in independent Brownian random environments. The competition rates depend on the spatial position and both population measures. Under suitable assumptions, we establish well-posedness for every pair of finite initial measures and derive moment estimates for the total masses. Existence is proved by constructing branching particle systems and establishing tightness of their empirical measure processes. Uniqueness in law follows from pathwise uniqueness for the system of stochastic partial differential equations satisfied by the associated distribution-function-valued processes.