Your challenge, with a POD of no earlier then 711 AD and no later then 1500, is to create an Islamic state with the great majority of its population being Muslims that controls all of the Iberian Peninsula that lasts until the modern day. Bonus points if the country is a Westernized liberal democracy. Even more bonus points if you can do it with as few changes in OTL history as possible (so that there is still a U.S in ATL, along with the EU, World Wars 1 and 2, Steve Jobs, etc.)
The last gets thrown right out the window; if we were talking about, say, New Zealand or Australia then you might have some tiny hope, but this is
Spain. There's no way to introduce a major change that early that won't butterfly away virtually every specific non-geological event more than a few centuries after the POD. For Europe, the butterflies have affected the entire continent within 100-150 years tops. So the extra bonus points are ASB.
Westernized liberal democracy...tricky, but doable. It's probably a long and ugly road getting there, though, much like it was for Spain in OTL.
As far as the survival of Al-Andalus...the biggest thing would be destroying the remaining Christian resistance in Asturias. In OTL, that region was never fully conquered, and was where the Reconquista was launched from. According to ever-helpful Wikipedia (sorry, but I don't have any better sources available at the moment), the Battle of Covadonga in 722 was what kept Asturias from falling the way the rest of Iberia had. Have it go the other way, with the Muslims winning, and there are no longer any Iberian Christian kingdoms left. Any Reconquista would have to start from across the Pyrenees.