Emily and Eliot both have eyes that ocassionally tear up like that and yes I have had both of them to the vet. The first time he gave me ointment but this last time he just brushed the whole thing off and told me it was a combination of tearduct and allergies and possibly they mark more vigorously on one side - but to watch for puffiness or increased discharge. So, it comes and goes and I just try not to worry about it.
Maybe since this is the first time for Corkscrew though - you should watch it extra carefully. I bet it goes away in a couple days if it is just allergy.