I've run into some problems with this surname, as in talking with my family there have been several different spellings. From what I understand, this is a fairly rare name. The versions I have seen include RET, RHET, RETT, and RHETT.
The language Furlan/Friulan is spoken in that region, so due to its influence there may even be a vowel at the end of the name that just never was pronounced.
Any and all help would be appreciated.