[cracks knuckles] I'm actually working on this as we speak.
The last POD is 1877, ant the Constantinople Conference intended to settle the "Eastern Question". The real power in the Ottoman Empire at the time was the reformist Grand Vizier Midhat Pasha, who was virulently against Western interference in Ottoman affairs. The Ottomans had managed to put down the revolts in Bosnia-Herzogovian and Bulgaria (the latter of which was foreign-incited) and handily defeated Serbia, but the Powers were demanding reforms in the "Christian" provinces. Midhat rejected all proposals as unnacceptable violations of Ottoman sovereignty (true, but beside the point). Sultan Abdul Hamid II was in favor of compromise and accommodation of the powers, but had just assumed the throne, had not consolidated his position, and was unable to override Midhat. As a result, the conference was a failure and provided a pretext fro Russia to declare war, even theough they didn't really have a casus belli, since even the Christian leaders were against Russian military intervention.
One of the reasons Midhat Pasha was so powerful was due to the assassination of the War Minister, Husseyn Avni Pasha, by an outraged soldier who blamed the reformists for the death of Sultan Abdul Aziz. But what if he had failed? Then there would be two powerful opposing ministers that the Sultan could have played against each other and got his way, accepting some of the European interference (which he could have subverted and undermined over time as he did historically), and there would have been no war.
The other serious problem the Ottomans had was financial. The reformists made a lot of progress modernizing the empire and its government, but they didn't really understand finance and accumulated horrendous debts. Abdul Hamid was able to consolidate all the loans and reduce payments to a reasonable sum, but the empire had lost a thrid of its revenues without really being able to lower expenditures, so the empire was essentially breaking even. With no war, the Ottomans have a significant amount of extra money to use for education, infrastructure, and the military.
So, there are the Balkans. Nicholas II was much more interested in East Asia, and maintained relatively friendly relations with the Ottomans, so once he's on the throne the Russian danger largely passes.
On to Africa. Tunis was promised to France as a quid pro quo for the British occupation of Cyprus in the Treaty of Berlin (which doesn't happen here). Thus, France can't snatch it without the threat of a general war. When troubles there reach crisis point, the Ottomans move in and restore control (Tunis was also had a huge debt load) and assume responsibilty for the Tunisian debt. Meanwhile the Urabist revolt in Egypt alarms the Porte, which fearing European intervention moves reoccupies Egypt, after negotiations with the Urabists provide for the appointment of Egyptians to the administration and military (prior to this controlled by the Turkish/Circassian/Albanian ruling elite and a large number of extremely highly paid Europeans). Subsequently troops are sent to restore order in the Sudan and end the Mahdist revolt (would have been easy in 1882-83).
In Arabia, in the early 1870s, the Ottomans had reasserted their control over the Persian gulf coast of the peninsula, and cotrolled the whole coast up to and including Qatar, and nominally the entire Arabian interior (the Saudis acknowledged Ottoman suzerainty historically until WWI). Given greater financial and military resources, the whole peninsula except Aden, Oman, the Trucial States, and Bahrain are integrated into the empire.
While agitation by the Balkan Christians remains an issue, the stonger ATL empire is able to contain the problem. Little known to most people, the Balkans were about 45% Muslim in 1876, and the Danube province, which formed the bulk of Bulgaria, actually had a Muslim majority.
In Central Africa, the Ottomans, in control of the Sudan and Libya, are able to enforce their claims to much of the Sahara region encompasing today's Chad and much of Niger (Historically, Bornu, the largest state in the area [around Lake Chad] raised the Ottoman flag in 1891 to try to avoid falling under British & French rule).
Eventually, a general European war occurs, say around the WWI timeframe, and the Ottomans stay neutral and take advantage of the opportunity to abolish the capitulations and extend and consolidate their power in the Mid East, assuming control over the entire Arabian Peninsula except Aden.
Migration and population growth swamp the Christian population of the Balkans, and oil development (Iraq was known to have oil since the 1880s) provides the empire with the financial resources to become a serious power. the collapse of Tsarist Russia allows the Ottomans to move into the Caucasus, Crimea, and Central Asia, liberating the Muslim populations and doubling the number of ethnic Turks in the empire.
By 2004 the population of the Ottoman Empire is 400 million, and it controls the vast majority of the world's oil reserves. Extremely low taxation and local autonomy hold together the enormous empire, as well as Muslim solidarity.
Here's a map of the empire in 1923 - ignore the blue lines; they are the routes for the Imperial Airship Scheme.
