Don't rat eyes constantly produce poryph, but just to keep the eyes moist, and then they over produce in times of stress? Maybe if there isn't an eye to keep moist it might just leak? especially if is a birth defect (or mini eye) rather than surgery? Don't the surgens sew the lids closed? I don't really know, but that was my first guess when I read this.
What I do know tho, is that a rat I re-homed has a leaky eye, constantly. She was on ABS for about 6 months with no improvement. I thought it would settle down when she got into a good home, but it didn't (she went with some friends of mine) She's perfectly healthy other wise, and I consulted a number of vets and different people who thought maybe its just her.