Tricky one, Mate...
The norm for this is to apply for a tourist visa ( 1 month initially ), then once in country extend it for upto to three months. After the three months is up, a new tourist visa and a day trip out of ther U.A.E. to validate it will be required.
British Passport holders have it easier with automatic 1 month tourist visa's on arrival then the above process for extensions.
The obviously ploy, is to enter the country on a torist visa then find a legitimate employer for your better half who is willing to provide her with a work visa.
It has to be said that "living together" in Dubai should not be a problem unless you fall foul of a local or a non-local spy which is not difficult in some circles and the reason I left Dubai with my fiancee firmly in tow.
It has been known for non-married couples who are caught alone together ( not neccessarily naked and obviously guilty of any "fornication outside wedlock" which is a criminal offence in the U.A.E. ) to be locked-up and/or lashed and/or deported in certain emirates and not that long ago, either. This is usually resrved for our collegues who originate east or south of the U.A.E. but a British couple were subjected to this as recently as 1996 if my memeory serves me well. They were caught naked and obviously "at it" and should have known better than to do this sort of thing in a densly muslim popoulated area of Sharjah.
Dubai takes a more liberal view, but do NOT **** THEM OFF for any reason. It's their country and they make the rules. Always remember that.
Respect them and their culture and your respect may be returned.
Whooaahh!!!