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Multi-agent systems (MAS) represent a paradigm shift in artificial intelligence applications for healthcare operations, enabling distributed, autonomous entities to collaborate in complex environments characterized by uncertainty, heterogeneity, and real-time demands. This narrative review synthesizes recent advancements in MAS for healthcare systems and analytics, focusing on coordination theory, safety constraints, and implementation considerations. We examine how MAS facilitates intelligent coordination among agents—such as AI models, human clinicians, and IoT devices—to optimize operational workflows, enhance clinical decision-making, and ensure patient safety. Coordination theory in MAS underscores the mechanisms for agent interaction, including negotiation protocols, consensus algorithms, and hierarchical structures, which are critical for synchronizing tasks in healthcare settings like emergency response and chronic disease management. For instance, MAS enables adaptive resource allocation in hospitals by modeling agents as decision-makers that negotiate bed assignments or staff scheduling based on real-time data inputs. Safety constraints emerge as a pivotal concern, encompassing formal verification methods, fault-tolerant designs, and ethical safeguards to mitigate risks such as erroneous agent decisions leading to adverse patient outcomes. Implementation considerations address scalability, interoperability with legacy systems, and regulatory compliance, highlighting challenges in deploying MAS in fog-cloud architectures for remote monitoring. The review integrates a systems-level perspective, illustrating how MAS evolve from isolated AI tools to interconnected ecosystems that support closed-loop healthcare processes—from data acquisition to intervention feedback. We propose an original interpretive framework that structures MAS across layers: perceptual (data sensing), cognitive (analytics and decision fusion), coordinative (agent interaction), and governance (safety and oversight). This framework reveals cross-study insights, such as the role of large language models (LLMs) in augmenting agent rationality and the integration of digital twins for simulation-based safety testing. Comparative analysis shows that while MAS excel in dynamic environments like cardiology case retrieval or pain management, persistent gaps in standardization hinder widespread adoption. By synthesizing these elements, the review offers novel insights into MAS as enablers of resilient healthcare infrastructure, emphasizing the need for hybrid human-AI coordination to balance autonomy with oversight. Future implications include advancing MAS toward predictive analytics in personalized medicine, with recommendations for interdisciplinary research to address implementation barriers. Ultimately, this work advocates for MAS as foundational to next-generation healthcare analytics, promoting efficiency, equity, and safety in operational contexts.