What did they call the realm of the gods in Sumer ?
Hello! I’m not a mythology expert, but as far as I know, there are three answers, and no answer, to your question.
Unlike in, say, ancient Greece, where the gods had a specific geographic point of residence (the summit of Mount Olympus), there was no such location in Mesopotamian myth. Instead, deities were generally associated with an, the sky, as the sky god Anu was king of the gods, and many other deities were his children or descendants. Anu is an odd deity, depicted rarely or never in art and featuring as only a minor character in myth, despite nominally being a or the supreme deity.
Individual gods and goddesses, however, were not considered to have their residence in the sky or some faraway location - they lived in their dedicated temples, which were seen as their houses (e) in addition to where they were to be worshipped. The central temple (egal, “big house”) of each city was the home of its resident deity, so people would travel from all over to e.g. Inanna’s temple at Uruk in order to literally visit her home, leave offerings and propitiate her. Other vocabulary would parallel as well, as with the unug, the holiest room at the core of a temple, which literally just means “living room”. This temple-as-residence applied to minor deities and guardian spirits, too, as an uzug shrine to a lamma protective spirit was also seen as that spirit’s home, and keeping it maintained meant the spirit would want to continue living there!
There are specifically supernatural realms in Sumerian myth, not visitable by us regular people, as in the Abzu, the great underground sea, or the underworld, which went by numerous names including Irkalla “big city”, Arali, Kur “mountain” or Kigal “great place”. Some deities in myth “die” and thus move to the underworld, but it wasn’t a regular residence for them. So these would be “realms of the gods”, but not in the sense you’re probably thinking.
I hope that answers your question!