Pasting my answer to a similar prompt:
edit: thanks for the feedback. Before the.... happening... this was one of my favorite subs. I want to see it like it was in its glory days, but until then I'll try and paint as good a picture I can.
TL;DR at the bottom.
The Byzantine Empire, assisted by the Catholic west in the First Crusade, successfully pushed the Turks out of Nicea and western Anatolia in OTL, but stopped short of retaking Anatolia as (it is hypothesized) reconquering Anatolia would mean giving central powers to local lords and authorities.
In ATL, the Byzantine emperor Alexios Komnenos utilizes the First Crusade to the best of his ability in order to retake Anatolia, drive the Seljuk turks out of the region, and establish dominance in Anatolia. The success of the crusade along with Alexios realizing the necessity of good relations with the rest of the Christian world reveals that if the Byzantines are to survive, they must cooperate with those in the best position to secure their territory.
Around 1100, the Byzantine Empire is secure. With Anatolia safe for now and healthy Byzantine relations with the Catholics in the west, the Byzantine empire is secure enough from outside threats to focus on the internal threats, abating the civil wars that ravaged the empire past the Fourth Crusade in OTL. As the crusades had two motivations (a. assist the weakening Byzantine empire as a form of Christian solidarity; b. reopen Jerusalem to Christian pilgrims), the success of part a allowed both factions to focus on part b. A combined force of Catholic and Byzantine and Christian zealots are inspired by their earlier successes and carve the Kingdom of Jerusalem, rabidly supported by all of Christendom, out of a region as far south as Judea and as far north as Antioch. The Kingdom of Jerusalem adds a perfect addition to the Byzantine counterweight to conquest-hungry Muslims.
With resources secured by the Byzantine empire consisting of what is now Greece and Turkey as well as the stability provided by healthy relations with the Catholic kingdoms, the further Crusades as we know them didn't happen, and Constantinople was never sacked by unrestful Crusaders during the Fourth Crusade.
While the Mongols invade and successfully destroy Kievan Rus in OTL, a healthy Byzantine Empire and a tradition of Christianity uniting for self-preservation brings about alternate "crusades" for the defense of the eastern steppes. Out of a sense of urgency, the Byzantine emperors over a period of 100 years reform the military and administrative structure of the empire, putting it on near-equal footing with its western counterparts. The newfound military strength and religious zeal allows the Europeans to expel the Mongols from what is now Russia/Ukraine, and as the Byzantines played no small part in the defense, they claim Crimea and the territories bordering the Black Sea in the north for themselves. As the entity of Russia hasn't yet begun to consolidate power, they do not resist these claims, and Byzantine authority is cemented until the 1600s.
With a friendly Orthodox state to the south, Russia has no Ottoman threat and instead focuses in the north on Sweden and the Commonwealth as well as the east into the Siberian wastelands. Byzantium has at this point firmly established its dominance of the Black Sea with direct control of the Anatolian coast and Crimea, and client kingdoms in Armenia, Georgia, and the rest of the territory on the western coast of the Black Sea.
With the lack of an enemy in the east, the Pope declares more crusades as late as 1600 in an attempt to return the African coast to righteous hands, and with the lack of Ottoman support the kingdoms of Morocco and Tunis are conquered and Christian rulers are installed. With the exception of what is now Egypt, the Mediterranean is once again "Christian", providing much-needed stability for the Byzantines.
Industrialization means great things for the Byzantine Empire, having learned its lessons about holding onto archaic traditions previously in the millenium. The Byzantines healthy relationship with the west keeps the emperor informed of technological revolutions in London, and the innovative Byzantine empire declares an alliance with England as they share technological developments and catapult each other to heights of production previously unreached.
The advent of the Enlightenment led to ideas of Enlightened Despotism, and the Byzantines of ATL were no exception; close ties with England as well as the stability of the empire allowed it to reform its government, changing from an absolute imperial authority to a constitutional monarchy complete with a full renovation of the ancient Roman laws as the basis for the constitution.
As we near World War 1 in OTL, tension in the Austrian domain reach a breaking point, resulting in the disassembly of the Ottomans. In ATL, an industrialized Greece and bountiful Anatolia place the Byzantines in a powerful position to stabilize the region with the help of the rest of Europe. The strong Russian allies in the south help put down the Bolshevik revolution, keeping the Russian imperial family in power and negating the threat of Communism to the west.
A fellow known as Hitler takes power in Germany, annexing left and right until the Allies of OTL with the addition of the Byzantines must take action. Italy engages with the Byzantines who, continuing with the theme of a top-tier military, beat back the Italians and occupy the land until their surrender in 1941. The Byzantines continue the occupation until the end of the war, which never gets as out-of-hand as in OTL as the Russians align with the Allies from the very beginning. Since there is no Communist-Democratic split of Germany, it is allowed to remain a whole nation.
It is increasingly apparent, however, that a monarchy is not the way of the future. With the Byzantine accomplishments in World War 1 and 2, the feeling of Romanitas (Roman-ness) is stronger than ever, and there are calls from within for the Byzantines to return to the glory days of Rome, albeit with a Greek twist on things. The occupation of Italy turns to annexation, and recognizing when things have run their course, the Byzantine emperor declares the end of the Byzantine Empire and the beginning of the New Roman Republic, complete with a system of representation similar to the US Senate and House.
Italian citizens, while fueled by a sense of nationalism, accept this new authority as the government is able to appeal to their pride in the long-gone Roman Empire. Out of gratitude for Byzantine aid in the war, the Allies accept the formation of the New Roman Republic.
With no Cold War to polarize the world, there is an era of peace and prosperity not known since the Pax Romana. In fact, the combined influence of America and the New Roman Republic lead the world into the 21st century.
TL;DR: Pax Romana Redux. Basically, my game of Crusader Kings 2 that continued into Europa Universalis 4.