How will divorce (or bigamy) be handled? For that matter, paternity?
Honest question. A similar idea is floating around on this side of the Mississippi.
We have ecclesiastical courts for divorces in the Orthodox Church and without permission from one of those to remarry (given only to the innocent party in a divorce undertaken for a cause recognized under Orthodox canon law), divorcees may not marry. How previous non-sacramental marriages terminated (by divorce or death) before one party to them is received into the Church are treated is an interesting matter of pastoral practice, with few good answers.