Does it really have to be 12 strings? What about 11 or 13 strings?
Swedish luthier Georg Bolin construced relatively successfull 11-string guitar that usually goes under name "alto-guitar". It is pretty much a lute in guitar shape sounding like a modern guitar rather than lute. I think there was 13 string version as well, but I am not sure about this. I also think it was quite popular during 70's at least in Sweden, I am not sure if the popularity still holds, but there is some "avantgarde" music written for it, as well as some arrs. Göran Söllscher is a well known guitarist who played quite a lot on that guitar, you can see him on youtube playing it. He recorded all Bachs lute works on it, and even some Beatles and other music. I can say those Bachs recordings are truly among best ones you can find. Carsten Gröndahl is another guitarist, this time from Denmark who used it on some recordings, I don't know if he plays it on concerts.