The distinction between a city and something else (like a proto-city or village) lies in the division of labor afforded by agricultural surplus. A "city" then would have full-time workers living in the interior (artisans, craftspersons, builders, etc.) with farmers on the margins supplying the food stores. Michael Balter, writing for the November 1998 issue of Science, notes that "For archaeologists, the difference between a village and a city is not just a matter of size but hinges on the social and economic relationships within a population. Thus the earliest cities in Mesopotamia—such as Uruk—were made possible by agricultural surpluses that allowed some people to quit farming and become full-time artisans, priests, or members of other professions. Meanwhile, the farmers who provided food for these urban centers continued to live in outlying villages." Balter goes on to note that no strong division of labor is indicated in the Çatalhöyük ruins. Residents appear to have done their own construction, built their own houses (often upon demolished parts of existing houses, thus accounting for the layers of ruins), created their own art (as opposed to relying upon full-time artists and artisans), and had no public buildings or monuments. Thus, Balter writes that Ian Hodder, the leader of the excavation, "has tentatively concluded that the basic social units at Çatalhöyük were extended families grouped together in clusters of four or five houses, which carried on their daily activities more or less autonomously.

Consequently, archaeologists view Çatalhöyük as more of a conglomeration of villages, or proto-city, rather than a full city such as what we see later in Mesopotamia.

I'm not sure what you mean or what assumptions you are placing within the phrase 'real city'.

Generally speaking, urban centres emerge in southern Mesopotamia in the 4th millennium BCE and are characterized by a suite of institutions, including hierarchical leadership and a professional bureaucracy. If this is what you are meaning with 'real city', then no Çatalhöyük isn't one.

The term proto-city is closer, but I would urge caution. Not because the term is wrong per se, but because this term could imply a linear progression from Çatalhöyük to later urban centres. Çatalhöyük is often studied in conjunction with other early large-scale settlements, such as Jericho. This is not because these settlements share a common origin or many similarities, each is quite unique and esoteric, but because they are so dissimilar from everything else that their individual suite of differences becomes a shared trait. These settlements are often collectively referred to as 'proto-cities', as they express some, but not all, of the features we see in later urban centres. Again, just as a precaution, this term does not mean that these earlier settlements were related to, or developed into, later urban centres.

I hope this answers your question.

TL / DR 'Real city' is an ambiguous term, but sites like Çatalhöyük do lack features found in later urban centres so it is generally not referred to as a city. Proto-city is a better term that is used in the literature, just be aware that it doesn't mean Çatalhöyük led to the development of later cities.