Provably correct distributed protocols, which are a critical component of modern distributed systems, are highly challenging to design and have often required decades of human effort. These protocols allow multiple agents to coordinate to come to a common agreement in an environment with uncertainty and failures. We formulate protocol design as a search problem over strategies in a game with imperfect information, and the desired correctness conditions are specified in Satisfiability Modulo Theories (SMT). However, standard methods for solving multi-agent games fail to learn correct protocols in this setting, even when the number of agents is small. We propose a learning framework, GGMS, which integrates a specialized variant of Monte Carlo Tree Search with a transformer-based action encoder, a global depth-first search to break out of local minima, and repeated feedback from a model checker. Protocols output by GGMS are verified correct via exhaustive model checking for all executions within the bounded setting. We further prove that, under mild assumptions, the search process is complete: if a correct protocol exists, GGMS will eventually find it. In experiments, we show that GGMS can learn correct protocols for larger settings than existing methods.