Joe asked me to comment on this thread, and then found the material above form email@example.com, which covers most of what I was going to add to this discussion. The tune is definitely an old sephardic tune, though when you think Judeo-Spanish, you should remember that a significant portion of the Jews thrown out of Spain in 1492 went to present-day Turkey, Greece and the Balkans, which may account somewhat for the oriental sound of the tune.
The language is Ladino, also called Judezmo, a mixture of old Spanish and Hebrew. Note the variations above: "Mi alma se escurese" versus "Mi neshama s'e scurese": the word for "soul" being "neshama" in Hebrew and "alma" in Spanish. Ladino uses "muestro" rather than "nuestro" for our. And "good-bye" is "adio" rather than "adios"--a tribute tot he Jewish concept of the singularity of G-d.
And lastly, when I sing the tune myself, I'm most likely to use it with the words for the table-song "Tzur Mishelo Achalnu".