"Rides herd" is an idiomatic expression that means to supervise or manage a group of people or things closely. There are several synonyms for this expression that can be used interchangeably. One such synonym is "keep a watchful eye," which conveys the idea of closely monitoring or overseeing a situation. Another option is "stay on top of," which implies the need to exert control and remain in charge. "Watch over" is yet another synonym that suggests the need for vigilance and attentiveness. Overall, these synonyms convey the idea of careful supervision and management, and can be used to replace "rides herd" in a variety of contexts.