There is a consensus though on what it is not: a universal jurisdiction, direct governance over other jurisdictions even if in exceptional cases, infallibility, and that it is necessary for salvation.
The debate is how exactly this primacy is to be exercised: merely in a honorary fashion or with some actual presiding prerrogatives and if any prerrogative, which ones.
My take on this, is that eventually people will acknowledge what is written in the Justinian institutes: that the "First Among Equals" is the "Head of the Bishops", that is, the presiding chair of the Synod. That is far more than the stricts conciliarists think and far less than radical papists advocate. Also that title is not theological or ecclesiological, but administrative. It is a rank, not an order. The extreme metaphysical dogmas related to the see of Rome necessarily being this presiding chair will have to be put aside. The criteria was clearly a mix of prestige from ecclesiastical history, social, political and economical importance and "spontaneous respect" due to firm defense of the Orthodox faith.
If Rome accepts the above, plus the revogation of the Filioque from the Creed (or if Rome and Orthodoxy can agree with the formulation "and through the Son"), and if Rome can acknowledge Immaculate Conception as a theologumen instead of dogma, then, IMO, union could happen immediately with Rome again as the first see. After all, today it *is* the most important see in social, political and economical importance, it does have a beautiful ecclesiastical history and if it confesses the Orthodox faith, it will comply with the three criteria.