Joel I. Barstad, Russian Catholic and professor of theology at St John Vianney Seminary in Denver, attempts to answer this question. From a rather interesting blog entitled The Augustana Greek Catholic: An Irregular Journal of Ecumenical Experiments.
Many Greek-Catholics define themselves as Orthodox-in-Communion-with-Rome and appeal to the First Christian Millennium as providing the foundation for this communion. This way of identifying themselves found confirmation in Joseph Ratzinger’s proposal that, with regard to the primacy, Rome need require nothing more from Orthodox churches than what was acknowledged during the First Millennium. With a similar conviction the Melkite synod in 1995 adopted the Zoghby Initiative as the framework within which it might reestablish communion with the Antiochian Orthodox Church, without breaking communion with Rome.
This article considers the viability of such a project in light of Pope John Paul II’s Ad tuendam fidem, and its companion commentary on the Profession of Faith and Oath of Fidelity, published in 1998.