Is there any problem in Jordan about travelling with a girlfriend (boyfriend) ?
To put it shortly, “No”. There is no difficulty whatsoever about sharing a room with a member of the opposite sex PROVIDED that they are not nationals of the country where you are. This is heavily frowned on and may be drawn to police attention. But somebody coming from outside ? No problem whatsoever (even a Jordanian in Syria or a Syrian in Jordan would not be a problem!). Most certainly two tourists raise not the slightest eyebrow. There is no need to wear a fake wedding ring, no need to talk about “my wife”, no need for the slightest pretence. This is quite simply considered to be nobody’s business but your own. Equally there is no need to “prove” a marriage if it actually exists! This holds good for the whole of the “touristic” Middle East.