The phrase "put tail on" refers to the act of following someone in a discreet and sneaky manner. Synonyms for this phrase can include "trail," "shadow," "track," "stake out," "monitor," "follow closely," and "keep a close watch." These words all imply a certain level of surveillance or observation, and may connote a sense of suspicion or mistrust. However, in some contexts, "put tail on" can simply be a lighthearted way of saying "keep tabs on," implying a desire to stay informed or involved in someone else's activities. Ultimately, the choice of synonym depends on the tone and intent of the speaker, as well as the context in which the phrase is used.