"Ride herd on" is a commonly used idiom that means to closely manage or supervise a situation or group. However, there are several other synonyms that can be used instead. For instance, one could "keep tabs on," "watch over," "monitor closely," "supervise," "control," "manage," "oversee," "manage closely," or "be in charge of." Each of these synonyms has slightly different nuances, but they all refer to the act of closely monitoring and managing a situation or individual. In general, the choice of synonym depends on the specific context in which it is being used.