Slimey eyes

Do her eyes ever seem red or sensitive?
"Dry eye" is not uncommon. It can be diagnosed with a tear test at the doctors office and is usually treated with a prescription oil.
You can safely use plain mineral oil, a few drops in each eye several times a day. If it seems to help then she may just have dry eyes.
That might explain why the Terramycin helps but never "cures" anything.
If that doesn't help after a few days you may see about getting some oral antibiotics, the green color may be a sign of infection.