The Holy See is a sovereign state, but the Pope is not just the head of state of the Vatican City; he is also the head of the Roman Catholic Church, and these two roles are separate and distinct. If tomorrow the Italian government decided to seize the Vatican this would not make a difference with regards to the Pope's status within the Church.
Moreover, the Catholic Church (of which I am a member( regards itself as divinely sanctioned and exclusively so among religious congregations; accordingly, she considers herself the only legitimate religious force in the world.
The consequences of this state of fact is thus: the Pope may engage the Holy See in diplomatic relations with a non-Catholic temporal state, but he has no authority whatsoever to engage the Church in diplomatic relations with what Catholicism affirms to be illegitimate religious forces. It would be analogous to the U.S. saying that it did not recognize the Republic of China (based in Taipei) as the legitimate government of China but setting up an embassy there anyway. This is the problem with attempting to engage in "dialogue with Islam."
Naturally, of course, an individual Catholic may engage in dialogue with an individual Muslim. But this is different from the CATHOLIC CHURCH engaging in dialogue with ISLAM. Doctrinally, the Catholic Church has absolutely nothing to learn from Islam and "Islam," defined as it is by rejection of certain fundamental Catholic teachings (notably the Incarnation) cannot benefit from Catholicism without destroying itself (affirming the truth of Catholicism would be tantamount to denying the validity of Islam).
So what were you trying to say?