|Later depiction of Benjamin of Tudela|
Is it possible, however, that a partial answer was provided by a 12th century traveler from Spain, a sort of Jewish Marco Polo who went seeking settlements of Jews all over the known world and recording details about their lifestyle?
Benjamin of Tudela (1130-1173) was born in the little town of Tudela, in Navarre in northern Spain. That sums up what we know of his early life. Around 1165 he undertook a journey east—whether his purpose was to visit the Holy Land, to truly map out Jewish settlements, or to make business connections, we don't know—and spent the next 8 years traveling and writing down the details of his travel and of the communities he met. As a 12th century snapshot of life in the Middle Ages—particularly Jewish life across Europe and the Middle East—his Masa'ot Binyamin (Travels of Benjamin) is invaluable to historians, especially Jewish historians, despite its occasional inaccuracies.
If he heard of any community of Jews, he visited them and asked about their traditions and numbers. And then there is this:
There are men of Israel in the land of Persia who say that in the mountains dwell four of the tribes of Israel, namely, the tribe of Dan, the tribe of Zevulun, the tribe of Asher, and the tribe of Naphtali. "They are governed by their own prince, Joseph the Levite. Among them are learned scholars. They sow and reap and go forth to war as far as the land of Cush, by way of the desert. They are in league with the Kofar-al-Turak, pagan tribesmen who worship the wind and live in the wilderness.
And also, in Arabia, he found a very large Jewish settlement:
These tribesmen are of the tribes Reuven and Gad, and the half-tribe of Menasseh. Their seat of government is a great city surrounded by the mountains of the North. The Jews of Kheibar have built many large fortified cities. The yoke of the gentiles is not upon them. They go forth to pillage and to capture booty in conjunction with the Arabs their neighbors.If you are interested in more, you can read a translation of his work at Project Gutenberg.