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Emerging Metaverse applications, designed to deliver highly interactive and immersive experiences that seamlessly blend physical reality and digital virtuality, are accelerating the need for distributed compute platforms with unprecedented storage, computation, and communication requirements. To this end, the integrated evolution of next-generation networks (e.g., 5G and beyond) and distributed cloud technologies (e.g., fog and mobile edge computing), have emerged as a promising paradigm to address the interaction- and resource-intensive nature of Metaverse applications. In this paper, we focus on the design of control policies for the joint orchestration of compute, caching, and communication (3C) resources in next-generation distributed cloud networks for the efficient delivery of Metaverse applications that require the real-time aggregation, processing, and distribution of multiple live media streams and pre-stored digital assets. We describe Metaverse applications via directed acyclic graphs able to model the combination of real-time stream-processing and content distribution pipelines. We design the first throughput-optimal control policy that coordinates joint decisions around (i) routing paths and processing locations for live data streams, together with (ii) cache selection and distribution paths for associated data objects. We then extend the proposed solution to include a max-throughput database placement policy and two efficient replacement policies. In addition, we characterize the network stability regions for all studied scenarios. Numerical results demonstrate the superior performance obtained via the novel multi-pipeline flow control and 3C resource orchestration mechanisms of the proposed policy, compared with state-of-the-art algorithms that lack full 3C integrated control.