My full mission tree Byzantium run!
Allies are Great Britain, Shun and Scandinavia (formed by Denmark), Bohemia is my PU and Munich is my vassal. People rivaling me: France, Spain, Prussia, Portugal.
Most fun run that I ever had.
I must admit that I restarted like 30 times before the game that finally worked out. Some restarts happened 30 days into the game, one happened 100 years into the game. In my opinion, contrary to popular belief, the most difficult part of a Byzantium run is not defeating the Ottomans, but what comes after that.
First: a small guide on how to survive to 1460 and thrive a bit:
After the initial war with Epirus, I defeated the Ottomans in two wars. I found out that if I allied Serbia, Theodoro and Trebizond, Ottos would not attack me first, and instead would attack one of the beyliks (or sometimes Trebizond). If they attack Trebizond, you should dishonor, but anyway, this is your golden chance. By this time you should have 3 heavies and 9 galleys, or at least be close to completing those ships. Barrage+assault Gallipoli (I've found you need something like 16k troops to do this, preferably throw some mercs in the mix) while Otto's troops are on Anatolia and you are good to go for this war. In this war you should get around 70WS, which will be enough to get a deal worth all of your cores and with a little luck some reps. During the war, be sure to rotate your stacks around Otto's european provinces (to loot and let the provinces recover their loot), and transfer occupations of Gallipoli and Thesalonika to Athens. Also, after the initial assault to Gallipoli, fire the mercenaries you hired till you are at force limit. With all of this measures, you should be getting around +2 ducats per month during the war, so you will be able to pay many of the loans you took to make this war possible.
After this first war, the hard part begins.
During the truce you can see if either Venice, Genoa, the Knights or Naples find themselves without strong allies, but it is difficult for this to happen. Not a big deal if you cannot attack them at this time. You should also pay your loans and save some money. You'll need it. Also, improve relations+ally Muscovy. You're now big enough to be a desirable ally for them.
As soon as the truce expires and the Ottos have their troops on Anatolia, declare on them for one of your claims in Europe. Fully occupy Ottoman Europe (if they have conquered Albania+some venetian provinces it's good news for you, more warscore). Unfortunately, this will only give you like 20 WS.
So you have to go big or go home.
By this time, you should be blockading the straits, and the Ottos should have placed some stacks ready for crossing. Hire mercs so that you have some 50% more troops than Ottos biggest stack on the straits, and get all the morale and disc bonuses that you are able to get (privileges, religious icon, advisor, etc.) Let Ottos stacks pass one by one, blockade their retreat and you will stackwipe them, even with the most pyrrhic of victories. After this, hurry and try to occupy all the provinces you have claims on, as the Mamluks are guaranteed to declare on the Ottos at this time.
In this war you should try to grab as much land as possible. From Anatolia, take the southernmost province you have a claim on (the mamluks consider it of critical importance, so they'll take it in their war. It will be harder to get it from Mamluks than it is to get it from the Ottos), all of Albania, Macedonia, and the rest take it as you wish. Ideally, take provinces so that no big empire can take the claims that you have on Ottos for themselves.
This is the end of the guide for a succesful start. From now on, this is what happened on my run, more dynamic strategies are needed and sincerely, at some moments I got really lucky.
After this, I had 3 giants at your frontiers: Poland/Commonwealth on the North, Austria-Hungary on the west, and Mamluks on the East.
I hurried to conquer the Balkans. Hungary didn't fall on the automatic PU under Austria, so Austria broke their alliance with them to pursue the PU. In the meantime, the Commonwealth destroyed them, and while they were busy, I also invaded, getting all their provinces in Serbia and Bosnia. After these wars, Austria PU'd them, but by this time, I had already allied Austria. I managed to get Dalmatian provinces that they had conquered from Venice by asking them to return cores to Dalmatia. After I finished the corresponding mission I stopped expanding in that direction for a bit.
I took some bulgarian provinces that the Commonwealth got from the Ottos in a war where Muscovy called me in, so I was able to unlock some claims on Naples. Those claims arrived just in time, as Naples had just become independient (thanks to pretender rebels), so they were ally-less. I took 90 WS worth of provinces, enough to form a small coalition of Italians+Ottos.
On the East, I allied Qara Qoyunlu, and together with Russia, we managed to destroy Mamluks, that had conquered/guaranteed most of Anatolia. Eventually, I stopped calling Russia and then QQ. QQ broke their alliance with me when I conquered Syria (that they critically desired) and eventually rivaled me. Then, I conquered Georgia from them.
I conquered the islands on the mission tree (most of which were Spain's) by having a superior navy and by allying France.
Eventually, Russia rivaled me, after the Commonwealth had been wiped out by me/Russia/Austria/Bohemia. The league war was triggered and Austria+me lost badly. I lost a couple of provinces in Crimea to Russia. After this war, I broke my alliance with Austria (they were no longer the emperor) and instead I allied Bohemia( the new emperor)+Denmark+France. I also allied Shun, which had been the victor in the Chinese civil war and that rivaled Russia.
Before I was able to re-ally France, Spain declared on me (and Portugal joined), for the reconquest of one of the Balearic Islands. Shun dishonored. As I said, I had the superior navy, so the ticking warscore went all way down, and even when they were much more powerful than me. Bohemia, Denmark and I kept them at bay on Europe, and in Africa the Iberians had occupied Egypt, but as Egypt was TC'd, it didn't amount to much warscore. The attrition, the battle casualties they suffered, their fully blockaded coast, all had their toll on Spain's war exhaustion and the warscore, so I managed to peace out for some money, when my manpower was about to hit 0.
I declared on Russia while they were only allied with Orthodox Cilli (whom I had force converted =) ) and recovered my lost Crimean provinces and more. After this, I declared war on the pope for the Rome state. An easy war, since the pope had no super strong allies.
Then, I was about to declare on Austria for my claims in Croatia/Carniola, fully conscious that I was about to break my alliance with Bohemia for this, but Bohemia's king died without heir, and I got the PU
No one contested it. I went to war with Austria and a smallish emperor, so it was an easy war.
Unfortunately, a coalition formed, so I could not expand to Venice and unlock the mission chain for Northern Italy (and the Pentarchy chain!). I waited some years, I went conquering on the middle East (where Russia had just relinquished their guarantee over QQ), on Northern Africa and Granada, and just waited for my manpower pool to fill. After the coalition dissolved, I promptly conquered Venice and surroundings (which were held by Trent).
As I had already converted the five seats of the pentarchy to Orthodoxy, I was inmediately able to claim the Rome Restored! mission. It was simply perfect. I suddenly became tier 4 defender of the faith as most of the Spanish/Portuguese colonies became orthodox, as well as many of the remaining catholic statelets in the HRE. France and Portugal went Orthodox, and even the Pope (!) went Orthodox. There is nothing more beautiful than an orthodox pope.
After this, there were some run-of-the-mill conquests in Northern Italy, the most difficult of whose was against Spain for their Florentine lands. France had a couple of Italian provinces (including Saluzzo, which had no other core but the French), so I had to break my alliance with them. During the truce, they went revolutionary, and by the time the truce ended, they had no allies at all. And also they never played the colonization game, so my war against France was the easiest war I've ever had against France, in all my games. I dismantled the revolution and got my two Italian provinces.
I wanted to clean my borders, specially with Russia, but I'm thinking it is much of a Hassle. Prussia formed a hugbox by allying with Russia and Spain, and every one of these countries had bigger armies than me. Actually, when I saw that on the ledger, I noticed that I was in quite a precarious situation. But oh, well. I got everything I wanted from this run. Time for the next challenge.