Question: Which building has been the home to the most religious organizations, Gül Mosque or Hagia Irene?

Context paragraphs:
Gül Mosque: Gül Mosque (Turkish: "Gül Camii" , meaning: "The Mosque of the Rose" in English) is a former Eastern Orthodox church in Istanbul, Turkey, converted into a mosque by the Ottomans.

Faith branding: Faith branding is the concept of branding religious organizations, leaders, or media programming, in the hope of penetrating a media-driven, consumer-oriented culture more effectively.  Faith branding treats faith as a product and attempts to apply the principles of marketing in order to "sell" the product.  Faith branding is a response to the challenge that religious organizations and leaders face regarding how to express their faith in a media-dominated culture.

Gnostic church: Gnostic church may refer to a variety of religious organizations which identify themselves with Gnosticism.  Various Gnostic religious organizations include:

Religious broadcasting: Religious broadcasting is broadcasting by religious organizations, usually with a religious message.  Many religious organizations have long recorded content such as sermons and lectures, and have moved into distributing content on their Internet websites.

Religious Organizations Law: The Religious Organizations Law (宗教団体法 , Shūkyō Dantai Hō ) was a Japanese law passed by the National Diet in 1939 and enacted in 1940.  The law gave the state authority control over religious organizations.  Following Japan's defeat in World War II, the Religious Organizations Law was repealed on December 28, 1945, and replaced by the "Religious Corporations Ordinance".

List of LGBT rights organizations: This is a list of LGBT rights organizations around the world.  For social and support groups or organizations affiliated with mainstream religious organizations, please see "List of LGBT-related organizations and conferences".  For organizations affiliated with political parties, please see "List of LGBT organizations that affiliate with political parties".

Hagia Irene: Hagia Irene or Hagia Eirene (Ancient Greek: Ἁγία Εἰρήνη , Byzantine ] , "Holy Peace", Turkish: "Aya İrini" ), sometimes known also as Saint Irene, is a Greek Eastern Orthodox church located in the outer courtyard of Topkapı Palace in Istanbul, Turkey.  It is one of the few churches in Istanbul that has not been converted into a mosque.  The Hagia Irene today operates as a museum and concert hall.

First Council of Constantinople: The First Council of Constantinople (Greek: Πρώτη σύνοδος της Κωνσταντινουπόλεως commonly known as Greek: Β΄ Οικουμενική , "Second Ecumenical"; Latin: "Concilium Constantinopolitanum Primum" or Latin: "Concilium Constantinopolitanum A" ) was a council of Christian bishops convened in Constantinople in AD 381 by the Roman Emperor Theodosius I.  This second ecumenical council, an effort to attain consensus in the church through an assembly representing all of Christendom, confirmed the Nicene Creed, expanding the doctrine thereof to produce the Niceno–Constantinopolitan Creed, and dealt with sundry other matters.  It met from May to July 381 in the Church of Hagia Irene and was affirmed as ecumenical in 451 at the Council of Chalcedon.

Ayakapı: Ayakapı (Turkish: ""The Gate of the Saint", "The holy gate"" ) (the toponym comes from the Turkish word "Aya", derived from pronunciation of the Greek word ἁγἰα, mean.  "female Saint" and the Turkish word "kapı", mean.  "gate") is a quarter of Istanbul, Turkey.  It is part of the district of Fatih, inside the walled city, and lies on the shore of the Golden Horn.  During the Byzantine era, it was named ta Dexiokratiana or ta Dexiokratous in Greek, after the houses owned here by a certain Dexiokrates.  Its modern name comes from a church dedicated to Saint Theodosia which, according to Petrus Gillius, stood near the gate.  In Ayakapı lies one of the most important surviving Byzantine buildings of the historical peninsula, the Gül Mosque.  Moreover, in 1582 the Ottoman architect Sinan built here a Turkish bath, the Ayakapı Hamamı.  This structure is currently used as a storage for timber.

Serbian Orthodox Diocese v. Milivojevich: Serbian Orthodox Diocese v. Milivojevich, 426 U. S. 696 (1976), is a United States Supreme Court case in which the Court held that the judicial determination of the Illinois Supreme Court violated the First Amendment and Fourteenth Amendments.  In matters of dispute within hierarchal religious organizations, the Establishment Clause precludes intervention by civil courts regarding internal disputes of church governance.  Per the Establishment Clause, decisions imposed by hierarchal religious organizations are binding in civil courts.

Supporting paragraphs: Gül Mosque Hagia Irene
Answer: Gül Mosque