Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
I frequently come across this concept in OO documents. It is mentioned that supposedly there is some form of "spiritual primacy" or "spiritual supremacy" (they might be two different things, but given that neither are really defined and are used in the same context, I don't know) in 3 different OO Sees within their respective realms, those being:
-Alexandria among the African OO (Coptic, Ethiopian, and Eritrean)
-Antioch among the West Syrian rite OO (Syrian and Malayali)
-Etchmiadzin among the Armenians
Yet there is never provided a definition for what exactly this reality is. Can someone explain to me what this means? How, for instance, the Coptic Pope is spiritually supreme over not only his own church but also over the two other autocephalous African OO churches?
(I'm assuming the concept and reality is the same for Alexandria, Antioch, and Etchmiadzin. However, if they are at all distinct, please explain.)
This concept of supremacy is in regards to regional/local ordination and infrastructural organization. In the Tewahedo Ethiopian Orthodox, up until 1959 the Ethiopian Church received its clerical ordination and administration directly from Alexandria and the Holy See of Saint Mark, as according to tradition and precedent established at the First Ecumenical Council. In socio-political terms, this was an excellent and effective strategy which allowed Ethiopian indigenous power struggles to be checked in their quest for popular mandate and political legitimacy, as the Egyptian clergy in Ethiopia had always been the Samuel-type Kingmakers. Traditionally all ordination, from deacons and monks to regional hegemons and bishops was performed annually by the Egyptian Abun who functioned as the metropolitan of Axum/Ethiopia and today is the Patriarch of the Ethiopian Orthodox Tewahedo Church, from the Holy See of Tekle Haymanot.
Still, the supremacy of Alexandria is recognized by the Patriarch in Ethiopia, His Holiness Abune Paulos, and His Holiness Pope Shenouda III in Egypt has resumed cordial relations with Ethiopia after a few decades hiatus during the Derg era.
The Patriarch, before 1959 was essentially the supreme landlord of all the Church land, and was responsible for technical schemes like budgets, tithe collecting, land maintenance and selling, and of course ecclesiastic issues and church matters of all variety. He was the head authority, sort of like the Supreme Court of the Church, but also the Chief Executive and Prime Minister of all Church matters.
Spiritual authority I suppose is derived from ordination, which the Patriarch is the source of, as according to the Apostolic Succession and laying of hands and anointing of the Apostles, which is received under the auspices of the Holy See of Saint Mark (for Africa), and so the Apostolic continuity is maintained from a tangible relationship. Remember the priests have been given the authority to perform the Divine Mysteries, including Baptism and Repentance/Absolution, so people very much felt that spiritual matters were in the authority of priests, who were in turn invested this authority, in the case of Ethiopia almost all priests and deacons received it directly from the laying of hands and anointing by the Egyptian Abun sent from Alexandria.
I can not honestly speak for other jurisdictions as you mentioned, but I can only assume it is similar as we are all Orthodox.