The Lute originated from the Middle Eastern Oud to which chromatic frets were later added, but many believe that a quite Eastern approach persisted. This has been a huge area of debate, as indicated here. Nobody knows for certain how mediaeval instruments were played, so there is a lot of room for interpretation. Styles vary from Fine Art to Folky, and from foursquare European to Silk Road fusion.
Medieval music was modal and harmonised with Unisons, Octaves, Fourths and Fifths but Thirds were considered discordant so I won't be playing any modern triad chords. After a lot of discussion I've decided to have it tuned DADG, a drone tuning which Jo suggested. There is obviously a coincidence here with DADGAD guitar tuning and the Zouk (GDAD).