Having a girlfriend in Abu Dhabi

The fact is that as per law, couples who are not married should not be living under the same roof or be alone in a room together.  That is an act for which you can legally be prosecuted - assumption being sex outside of marriage or intent to do the same.

There is no law about having or not having a girlfriend, hanging out with them in public and so on i.e. the law only makes a distinction about being alone with the opposite gender in private. 

Now having said the above, practically speaking, no one cares.   There are a lot of couples who live together and are not married.  As long as you don't attract attention to yourself or create problems, no one will bother you.  Usually, the only way people land in trouble is through stupidity e.g. someone will throw a wild party, a neighbour lodges a noise complaint, police turns up, they misbehave with the police and the book is thrown at them.  Or public displays of "intense" affection, asked to stop, don't stop and a local complains.