OK: this one is basically L Neil Smith's "Crystal Empire", incorporating some bit of "The Gate of Worlds" and a little more plausibility.
A mutated version of the Black Death arises in France: it doesn't travel as well as the regular kind, so it doesn't spread too far, but western Europe gets a double dose of death, allowing for an easier Ottoman expansion and an effort by north African Muslims to do a re-reconquista (or would that be a de-conquista?) by way of Granada. European states most distant from the "front lines" manage to rally, but are in the end unable to prevent Islam from overrunning far more of Europe than OTL. The rather overextended Ottomans, meanwhile, eventually settle for ruling much of their vast conquests as puppets, while the northernmost parts of europe manage to slip out of their influence.
History happens. Due to butterflies from different events in the west, a Turkish *Timurid Empire manages to conquer the north of a less successful Ming dynasty, and over the course of several centuries of warfare, the south and finally Japan, benefitting from technological and military improvements filtering in from the Ottomans. The South Chinese, who like the Sung before them are pushed to look to overseas trade as a substitute for the lost northern lands, discover the route to the Americas, and as the Islamic Khanate slowly grinds south, many flee to the Americas for refuge, followed later by Japanese. The Scandinavians discover their own route to the Americas by way of Greenlands, and are followed by the Ottomans and their Moroccan vassals, who establish themselves to the south.
In the year 2014, the world is divided between three great empires, two of them Muslims: the Christian League of the North likes to think of itself as another big power, but it's not taken too seriously. Absorbing much European culture and philosophy, the Ottoman state has developed a primitive scientific community, and has made great advances in rule-of-thumb engineering: railways are binding the diverse regions of the Empire together, steamships spin their giant paddle wheels, a telegraph has been developed in spite of the still rather confused nature of electrical theory, and an industrial revolution of sorts has been going on for a while, if moderated by a paternalistic Ottoman state which doesn't stand for the sort of abuses normal in early British industrialization. The Empire is rather decentralized (although railways are starting to lessen that) and fairly tolerant of religious minorities, the tolerance that comes from largely unquestioned mastery - although that mastery is now under challenge.
The former Khanate is now the Holy Caliphate of the Central Realm, under a new southern dynasty which burn with the fervor of recent converts. They have no patience for the tolerant and easy-going nature of the Ottomans,  and with millennia of Chinese centralism behind them, are making a cohesive and centralized state out their Empire. The Emperor has taken the title of Caliph, claiming to have a better claim to it than the "degenerates" in Constantinople, and military planners are drawing up enthusiastic schemes for campaigns in disputed India, the islands to the south, or even Iran. All the world must acknowledge the Central Realms leadership of Islam! Their technology is derivative of Ottoman models and generally inferior, but they feel enthusiasm, good organization, and numbers will make up for that in case the Ottomans dispute their status.
It's not just rivalry with the Ottomans, though: it's partially sour grapes and a way of washing the sour taste of failure out of the collective national mouth after the attempt to invade the *Americas to regain control of their "disloyal subjects" ended up with their fleet sent to the bottom of the Pacific by the Threefold Realm's superior naval gunnery.
The Threefold Realm is three kingdoms or "Thrones" founded by Chinese, Japanese and some Korean immigrant mixing with native American societies, and have only been unified in one overarching empire for the last century and a half. There are substantial regional differences: for instances, the more solidly Asian northern "Throne of the Golden Mountain" doesn't cotton to human sacrifice to the extent of the southerners, although it has begun to catch on as a mark of Imperial solidarity.
(It's not quite Mesoamerican or Andean sacrifice: it's not specifically meant to raise the sun or feed the Gods, who can feed themselves, but meant to help maintain the Tao, the harmony, of Earth and heaven, and represents the willingness of the People to sacrifice themselves for the common good: victims are - at least in theory - meant to be willing and honored. It also doesn't take place in quite such job lots: the elaboration of ceremony is more important than poundage of heart-meats).
They are generally tolerant of polytheists who don't follow their odd mix of Taoism, Buddhism, and local faiths, but when it comes to the Muslim enemy no follower of Islam is allowed in their lands, and Christians and Jews, as "people of the Book", are usually confined to trading ports and border towns. They have created an extensive network of native vassals to serve as a "cordon sanitaire" between then and the arrivals from across the Atlantic, and are quite concerned with reports of conversions to Islam and Christianity in some of their more far-flung vassals.
The overall society is aristocratic, highly literate, and artistic, and while culturally quite divergent from old China consider themselves the "true heirs" to Chinese civilization: they hope to one day return and liberate their homeland (there is considerable dispute as to whether the locals can be "deprogrammed" or whether they will simply have to kill off most of the population and replace them with immigrants from the *Americas).
This is not entirely wishful thinking. The inhabitants of the Threefold Realm of the Dawn Lands are technically very adept, and their natural philosophy is in some ways more advanced than that of the Ottomans: of late, some of their deepest thinkers have begun to wonder what it might look like if one could ride atop a beam of light...
For now, they wait and watch the increasing tensions between the Ottomans and the Central Realmers. If a World War breaks out, the opportunities may be vast.
 They actually don't mind too much People of the Book, Jews and Christians, which in any event are few in number in the Central Realm. It's tolerance of _heathens_ like Buddhists and Hundus which really honks them off, an attitude which began to develop when China went Muslim-majority and treating "pagans" nice became less necessary.