I agree with Teri on using a type of fish oil to lessen the tear flow. I know with Snickie her eyes are a little more runny at this time of the year, as we spend more time outside, I know she suffers from some seasonal allergies. If we spend more time indoors, there is less tearing.
Several years ago she was diagnosed with conjunctivitis, months of expensive meds with no cure, her eyes ran like crazy with more tear staining. I finally asked the Vet to do the dry eye test and she was borderline, her eyes at that time were only producing 12 drops a minute, should have been at least 15, around 20 is ideal. Sounds strange with less tears being produced that her eyes would be more runny, huh. Anyways I did a lot of research and found that dogs were usually lacking oil, which they need to mix with the water and mucus in their eyes to be balanced, inadequate oil and their eyes run water. So I added more veggies and fish oil, and other supplements and 8 weeks later her tears were at the ideal.
Being a red\buff and a BNR, with light eyes and rims, runny eyes and tear staining is very distracting. With filtered water and oil added to her diet, her tears and tear staining has been eliminated.