I understood you this way: If you went to communion on a fast-day in the morning, you would break your fasting according to your tradition, right?
Well, if one goes to Communion on a fasting day, the fast is broken after Communion: you can eat and drink afterwards (in fact, you must), but the abstinence will continue, however, in most cases.
So, for instance, on the Wednesday of our three-day fast of the Ninevites, we celebrate a Liturgy after the Ninth Hour instead of after the Sixth Hour, and it follows the Lenten order. After the Liturgy, we eat, but our food is Lenten because the abstinence is not broken until after the next day's Liturgy. But let's say 29 June is a Friday this year: we would celebrate the Liturgy for Ss Peter and Paul, and because that feast trumps Friday, all foods are allowed. Basically, if a major feast falls within a fasting period
, we break the fast after Liturgy but keep the abstinence, but if it falls outside of a fasting period but on a fasting day
(W or F), the major feast outranks the fasting day and we break both fast and abstinence.
I don't know how the abstinence rules work in the other OO traditions, but I'm pretty sure that all of them agree that, after the Liturgy is celebrated, there is no fasting.
EDIT: I should add, since I noticed I forgot to address the "morning" part of your comment, that there is some "cheating" involved. During Great Lent, we are not allowed to eat or drink until after the Ninth Hour. Normally, the morning office precedes the celebration of the Liturgy, and this is composed of Matins, Third, and Sixth Hours. If a parish celebrates Liturgy in the morning on a day when the Liturgy is supposed to be celebrated after the Ninth Hour, then the morning office includes the Ninth Hour as well. So, on a fasting day, you could theoretically end the Liturgy by 9am, but "liturgically" you have already passed 3pm. It's one of those liturgical fictions that have been widely accepted.