What could be the cause of constant tearing in one eye but not the other?

Blocked tear duct: The most common reason for constant tearing (epiphora) is a blocked tear duct. The tearing is usually asymmetric - worse in one eye than the other. Warm compresses to the inner corner of the eyelid (next to the nose) along with a mild topical steroid and antibiotic drops can help the condition. If these measures are not successful a surgical procedure (dcr) may be necessary.