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The Canadian Hydrogen Intensity Mapping Experiment (CHIME) has emerged as the prime telescope for detecting fast radio bursts (FRBs). CHIME/FRB Outriggers will be a dedicated very-long-baseline interferometry (VLBI) instrument consisting of outrigger telescopes at continental baselines working with CHIME and its specialized real-time transient-search backend (CHIME/FRB) to detect and localize FRBs with 50 mas precision. In this paper, we present a minimally invasive clock stabilization system that effectively transfers the CHIME digital backend reference clock from its original GPS-disciplined ovenized crystal oscillator to a passive hydrogen maser. This enables us to combine the long-term stability and absolute time tagging of the GPS clock with the short- and intermediate-term stability of the maser to reduce the clock timing errors between VLBI calibration observations. We validate the system with VLBI-style observations of Cygnus A over a 400 m baseline between CHIME and the CHIME Pathfinder, demonstrating agreement between sky-based and maser-based timing measurements at the 30 ps rms level on timescales ranging from one minute to up to nine days, and meeting the stability requirements for CHIME/FRB Outriggers. In addition, we present an alternate reference clock solution for outrigger stations that lack the infrastructure to support a passive hydrogen maser.