I heard mixing Apple Cider Vinegar with their water or food can clear it up (you'll have to look online for dosage). It works by changing the acidity of the tears so that the red yeast and bacteria that causes the actual stain can't survive in the moisture on their face.
I chose not to do it because I don't like the idea of changing body chemistry, I'm afraid it could affect urinary health. But there is no proof of this that I could find, so I don't know if it's really something to worry about. I don't want to risk anything though because it's just an aesthetic problem.
I think applying it directly to the tear stain would work as well, but if you got it in their eye it would hurt pretty bad I'd think, so it's not recommended. I did it a few times and never got it in his eyes, but he HATED the smell and started avoiding me. I didn't want to cause him to have face touching sensitivities so I stopped.