We know that bastards born in Westeros are named Snow, Pyke, Sand, Waters etc.
But what is a bastard born in Essos named?
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Unlike Westeros, Essos is a heterogeneous continent of ethnicities, languages, and cultures. As such, it does not have a unified standard for treatment of extramarital children (or anyone else).
The bastard names you're referring to are in replacement for family names, so children outside marriage don't bring shame on the family.
In both the book and the series it's suggested that only parts of Westeros, not essos use family names. (The wildings are natives of Westeros and don't use Family names)
In Essos most folks don't even seem to know Jorah Mormont has a family name and call him Jorah the Andal.
Note the conversation between Shae and Varys
Varys: If you let yourself believe that a foreign girl with no name could spend her life with the son of Tywin Lannister...
Shae: I have a name.
Varys: You have one name. As do I. Here only the family name matters.
Update: In direct answer to the question, What is a bastard in Essos called? By his or her given name without indication of birth in or out of wedlock.
It is unknown for certain; but it seems that it is nothing special.
As Geroge R. R. Martin says, in essence: Only high-born children get a "bastard" name.
I'll quote from my answer there:
Bastard names are given only to bastards with at least one parent of high birth. So the bastard child of two peasants would have no surname at all.
Thus a bastard name like "Snow" or "Rivers" is simultaneously a stigma and a mark of distinction. The whole thing with bastard names is custom, not law.
So you can see that it is customary for natural children to receive the "bastard" name; and given that it is a custom of Westeros, it stands to reason that only "Westerosi" bastards receive the regional surnames.
The only notable mention of a non-Westerosi having natural children is Hizdahr Zo Loraq - who has fathered two natural children. But we do not know what their last names are.
Though I have never wed, I have two natural children, a boy and a girl, ....
[Hizdahr from A Dance With Dragons]