The Moorish territories in the Iberian Peninsula survive until the Age of Discoveries, where they become a front runner. Although the Empire has fallen in the last hundred years, Al-Andalus is today an important Republic within the European Economic Community.

Broad timeline

After a civil war, Muhammad III becomes Caliph of Cordoba with fully support of the Umayyads. After a revolt in the capital city of Cordoba (Qurtuba), Mohammed resettled in Seville (Isbiliya) and consolidates its power.
During the civil war, the kingdoms of Leon and Castille made some advancements into the Moorish territory and Zaragoza (Saraqusta) and other territories had became an independent kingdoms.
Mohammad III reconquers most taifa (rebel) kingdoms but Saragoza, with the help of Arabs and Berbers from Northern Africa. By this time, this state is usually known as the Caliphate of Seville.
The Caliphate of Seville waves war against the Almoravid while still in constant warfare from Leon, Castille and the Franks. The kingdom of Saragoza has moved into friendlier terms.
Seville finishes the conquest of the Almoravids, and secures the northern frontier against the Christian kingdoms. The Caliphate includes most of the Iberian peninsula and a great chunk of northwestern Africa.
Castille and Zaragoza wave war against Seville. Leon reminds neutral.
Seville is defeated in the Peninsula but hold her African possessions. During the war Castille has become a de facto protectorate of Zaragoza. The king of Zaragoza refutes the title of Caliph
The elders of Zaragoza declare that the Cathars are people of the book and therefore allowed to settle in the kingdom as longer as they paid their jizya.
The Albigensian Crusade happened on schedule and many Cathars were admitted in Zaragoza.
Several battles fought between Zaragoza and the Kingdom of France after the fall of the kingdom of Barcelona during the Albigensian Crusade. The French and the Crusaders manage to reach the city of Zaragoza by 1222 but are repelled.
The king of Zaragoza chooses Valencia as his capital. The kingdom is now mostly known as Andalus. Andalus also secures the Balearic islands and the Gibraltar Strait.

The Mongol expansion happens on schedule. As the eastern Muslim states fall many Arabs relocate in Andalus.

A Turkish *Ottoman empire raises on schedule and soon gets control of Anatolia, Egypt and great deals of Arabia. Many Arabs keep relocating in Andalus, and an Andalusian-Turkish rivalry arises in the Mediterranean.

Although many European wars involve Christian fighting the Muslims, there are also many wars between Christian kingdoms in which one of them would ally to either the Andalus or the Turks.

Governor Hakalbit, from the Andalusian province of Silves sends an expedition to take back Ceuta (which has been in Berber hands for the last century). After the success, Hakalbit sponsors exploration voyages, mostly to the Mediterranean and Atlantic coasts of Africa. This soon become a navigation school improving research on navigation techniques.
The Turks take Constantinople.
An Andalusian expedition reach *Cape of Good Hope.

by 1485 Andalusian sailors had reached India, bringing goods into Europe.

A Chinese expedition following Andalusian charts reach the cape of Good Hope and continues westward. They reach South America several miles south of Rio de La Plata and continue southward until OTL Magellan Strait. By 1497 they reach China after completing the first documented circumnavigation. Given the political turmoil, however, the Chinese didn't invest in further blue water expeditions for several decades.
John Ferris, a French sailor sponsored by the English monarchy, reach Newfoundland. It is believed that Navarrian fishermen regularly fish in those waters (and probably had already reached land) but there is no record of those expeditions, although Ferris had lived in Navarra for several years before moving to England.
Unaware of Chinese and the Ferris expeditions, an Andalusian fleet returning from India finds unfavorable winds after they left the Cape of Good Hope and is pushed to South American shore. They call it Bonaventura (good Fortune in Mozzarabic). After a couple of month reparing the ships, they return back to Africa and Andalus.

Some further expeditions by England and Andalus let them explore the Atlantic shores of North and South America. By 1510 other European kingdoms had organized or were organizing their own expeditions, mainly Leon, Navarre, France, Netherlands, Scotland and Denmark.

Corruption in Rome led to a renovation movement in Germany and France. A complete mishandle by Rome led for renovation to spread. Soon several Christian kings understood that they could use renewed national churches to cement their power. By 1550, no Christian kingdom (but the Papal States) cared about a unified church in Rome.
An Andalus expedition discovers the Magellan trait and reach the Pacific shore of South America. By this time, the Andalusians had founded several coastal cities in Africa and South America, although they had not participate in colonization attempts.
Andalus explorers reach the Tawantinsuyo (Inca Empire) and they are surprised how easily they could lot on that rich kingdom.
News on the lot of the Tawantinsuyo and on further legends on golden kingdoms reach back home in Andalus. This spread further expeditions with conquest in mind.

The English in North America had also founded commercial outposts, and the French had already conquered some Caribbean Islands and have permanent bases in Borinquen (Puerto Rico).

And English expedition from OTL Georgia reached the Mississippi, arriving to the Gulf of Mexico in 1512, where they built some ships surrounded Florida and reaching Cuba before they reached back their original port in *Georgia in early 1513.

Disease took a toll on Native American populations spreading from Peru and the Mississippi. This further helped European conquest.

The conquering of Peru (Tawantinsuyo) was completed.
The Panama Isthmus was crossed by Andalusians from the Pacific to the Caribbean.
Andalus conquered the Aztec kingdom.
Vienna fell to the Turks.
The Turks are defeated at Brno.
The Turks are defeated at Passau.

By 1550, Andalus had defeated and conquered all higher civilizations in OTL Mexico and South America. The English had also developed colonies in North America. Smaller though.

The French completed the conquest of Borinquen, OTL Hispaniola, Jamaica and Cuba.

Native Americans were not recognized as peoples of the book, so the Andalusians felt free to enslave any Native American who didn't convert to Islam.

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