The phrase "put a tail on" typically refers to the act of surveilling or following someone discreetly. There are several synonymous phrases which convey a similar meaning, such as "shadowing," "tailing," "tracking," or "trailing" someone. Another possible phrase is "keep tabs on," which suggests monitoring someone's movements and activities. Some less common terms for this activity include "stalking," "creeping," and "lurking." Each of these phrases conveys the idea of keeping a careful eye on someone without being detected, often with the intention of obtaining information or uncovering secrets. Regardless of the specific wording, "putting a tail on" someone is typically done with the utmost discretion and caution.