Christ is with us!
In my Orthodox jurisdiction, the person seeking the annulment/Church divorce goes through counseling with his parish priest and the priest writes a document that explains why the divorce/annulment is being requested.
If it meets the criteria of the 7 permitted reasons for the divorce (adultery, inability to have children, abandonment, homosexuality, imprisonment, heresy/departure from true belief,or both spouses desire to enter monastic life) the paperwork is forwarded to the diocesian Bishop. As far as I understand it, the Bishop and the Bishop alone determined the need for economia to permit the divorce, there is no formal marriage tribunal as there is in the Byzantine Catholic Churches or the Roman Catholic Churches.
The couple is required to go through pastoral counseling to determine if the cause for the divorce can be resolved without the divorce being finalized. In various orthodox jurisdictions there may be a period of repentence (I have been advised by those I know, who have had a divorce ,that this is 6 months to 12 months) required prior to coming back into full communion with the church (depending upon the reason for the Divorce).
Once the person has gone through this period, they may seek remarriage but the marriage ceremony after the first marriage is one with a very penitential tone and should not have the crowning performed (according to the Hapgood Service Book). It is interesting to note that even if one has not been divorced but is widowed the same marital requirements apply for the second, even the third remarriage. After the third marriage, the canons note that the person has been called by God to the single state and many of them are encouraged to enter into the monastery if they have no familial demands.
Your brother in Christ,
Thomas