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Fog Computing is now emerging as the dominating paradigm bridging the compute and connectivity gap between sensing devices (a.k.a. "things") and latency-sensitive services. However, as fog deployments scale by accumulating numerous devices interconnected over highly dynamic and volatile network fabrics, the need for self-configuration and self-healing in the presence of failures is more evident now than ever. Using the prevailing methodology of self-stabilization, we propose a fault-tolerant framework for distributed control planes that enables fog services to cope and recover from a very broad fault model. Specifically, our model considers network uncertainties, packet drops, node fail-stop failures, and violations of the assumptions according to which the system was designed to operate, such as an arbitrary corruption of the system state. Our self-stabilizing algorithms guarantee automatic recovery within a constant number of communication rounds without the need for external (human) intervention. To showcase the framework's effectiveness, the correctness proof of the proposed self-stabilizing algorithmic process is accompanied by a comprehensive evaluation featuring an open and reproducible testbed utilizing real-world data from the intelligent transportation domain. Results show that our framework ensures a fog ecosystem recovery from faults in constant time, analytics are computed correctly, while the overhead to the system's control plane scales linearly towards the IoT load.