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Robust pulse synchronization is fundamental in constructing reliable synchronous applications in wired and wireless distributed systems. In wired systems, self-stabilizing Byzantine pulse synchronization aims for synchronizing fault-prone distributed components with arbitrary initial states in bounded-delay message-passing systems. In wireless systems, fault-tolerant synchronization of pulse-coupled oscillators is also built for a similar goal but often works under specific system restrictions, such as low computation power, low message complexity, and anonymous physical pulses whose senders cannot be identified by the receivers. These restrictions often prevent us from constructing high-reliable wireless synchronous applications. This paper tries to break barriers between bounded-delay message-passing systems and classical pulse-coupled oscillators by introducing a new fault-tolerant synchronization scheme for the so-called anonymous bounded-delay pulsing systems in the presence of indeterministic communication delays and inconsistent faults. For low computation power and low message complexity, instead of involving in consensus-based primitives, the proposed synchronization scheme integrates the so-called discrete mean-fields, planar random walks, and some additional easy operations in utilizing only sparsely generated anonymous pulses. For fault-tolerance, we show that a square-root number of faulty oscillators can be tolerated by utilizing planar random walks in anonymous pulse synchronization. For self-stabilization, we show that the stabilization can be reached in an expected constant number of observing windows in anonymous bounded-delay pulsing systems with the pulsing-frequency restriction.