The phrase "kept tabs on" implies monitoring or keeping an eye on something or someone. Some antonyms for this phrase could include ignoring, neglecting, dismissing, abandoning, disregarding, or overlooking. These words suggest a lack of attention, interest, or concern. Another way to express the opposite of "keeping tabs on" could be to say that one is giving someone or something complete freedom or autonomy, trusting them to take care of themselves without any supervision or intervention. Whatever the opposite may be, it is clear that the phrase "kept tabs on" suggests a level of accountability and responsibility towards the subject at hand.