The expression "kept tabs" is usually used to refer to keeping a record or monitoring someone's activities. The antonyms for this could be "ignored," "neglected," or "abandoned," which all imply a lack of attention or interest. Alternatively, "trust," "freed," or "released" could also be used as antonyms, indicating a non-invasive or non-restrictive approach. These opposite ideas suggest that instead of keeping tabs on someone, one could choose to give them freedom and trust, or simply not pay attention to their actions. Ultimately, the choice of antonym depends on the context and intention of the original statement.