During the Middle Ages did the Eastern Orthodox Church "rule over" the Western Church at any time and if so, what were the issues involved?
The historical period known as the Middle Ages, more aptly called Late Antiquity, was a period of tradition and stability. While there were wars, conflicts was rarely between established nations under the auspices of the Catholic (Western) Church and instead between feudal lords or rival clans from the Nordic lands, such as the Normans. While the Church was largely unquestioned in religious affairs, there was the Eastern Orthodox Church that challenged the hearts and minds of the faithful, even if it was only a nameless fear rather than through outright conversion efforts. Despite one Church never claiming dominion over the other, as there were plenty of sects within Europe itself, such as the Cathars to do that, there was the belief that each tried to undermine each other was ever constant. After all, the easiest way to rally supporters is to see enemies all around you.
The most noteworthy breakdown was the Great Schism between the two that escalated minor differences with the breaking of communion in the 11th century. I would see it as an eventuality similar to early ecumenical councils (Ephesus) in which different beliefs were already being rejected (Nestorianism and Arianism) despite the relative youth of the Christian religion itself. At issue (among others) was the papal supremacy of the Catholic vicar and the place of Constantinople among the holy church cities throughout the church's sphere of influence. These disputes are usually placed at the behest of individual legates or church officials, as reconciliation efforts were attempted every few decades (The Fourth Crusade likely didn't do these efforts any favors) with most recent efforts at inter-religious communion being much more successful, largely by focusing on the similarities of their beliefs rather than differences.