Modern engineered systems increasingly involve complex sociotechnical environments where multiple agents, including humans and the emerging paradigm of agentic AI powered by large language models, must navigate social dilemmas that pit individual interests against collective welfare. As engineered systems evolve toward multi-agent architectures with autonomous LLM-based agents, traditional governance approaches using static rules or fixed network structures fail to address the dynamic uncertainties inherent in real-world operations. This paper presents a novel framework that integrates adaptive governance mechanisms directly into the design of sociotechnical systems through a unique separation of agent interaction networks from information flow networks. We introduce a system comprising strategic LLM-based system agents that engage in repeated interactions and a reinforcement learning-based governing agent that dynamically modulates information transparency. Unlike conventional approaches that require direct structural interventions or payoff modifications, our framework preserves agent autonomy while promoting cooperation through adaptive information governance. The governing agent learns to strategically adjust information disclosure at each timestep, determining what contextual or historical information each system agent can access. Experimental results demonstrate that this RL-based governance significantly enhances cooperation compared to static information-sharing baselines.