The premise of this question: The Komnenoi restoration manages to more or less reestablish the borders of the empire as they were at the end of Basil II's reign by the mid 13th century. Naturally some areas are more secure than others, and this does not mean that the interior of Anatolia isn't Turkified to a strong extent.
So would it likely seek to take and/or control the same areas (more or less) as the Ottomans did OTL? Not necessarily right away, but would that be a goal for an ambitious emperor? Would it be more concerned with the East? (the way the Western European powers were in the 16th century and beyond)? How would the Latin powers of Europe see it (as opposed to how the Ottomans got a very bad name as the Terrible Turks)?
Basically, how would it be like (and unlike) the Ottoman state regarding strengths, weaknesses, enemies and ambitions.
It seems that it would probably be different in more ways than just having the Hagia Sophia as a church, but beyond that...well, hopefully there are resources out there worth reading to get an idea of how this could have looked. Unfortunately, just reading a history of the Byzantines doesn't give a very clear picture of what could have happened had things not gone wrong from 1180 (or earlier) on.
Well, there are some thing you need to consider - and let me first start by saying I'm not really up to speed on Byzantium.
-The Crusades did a number on Byzantium in numerous ways.
-The Turkic interruption of the Silk Road hurt trade.
-Mongols and other such (*cough* Turks *cough*) continually eroded the Byzantines.
Now. Let's step back and look at the issue at hand.
You're suggesting that between the Komnenoi Restored (c.1080-1180), the the Byzantines might be able to retake most of Eastern Anatolia, Cyprus, some more of the Balkans, MidEast, and Apulia (S/SE Italy).
Not saying it is impossible, but let's look at the greater political scheme. The Great Schism did a number on the Byzantines. Now they had to deal with more direct Catholic attacks from the West AND the Muslim *(& Pagan) barbarians, Turks, and so forth from the East.
There's really little wonder that within just a couple of decades or so after the Komnenoi fell out of favor (what with civil war and all) the Latin Empire was established by power-hungry crusaders from the west. Not only that, but after a couple centuries of warfare in the Levant, the smaller Muslim states were being eaten by Christians who were in turn eaten by the bigger more centralized Muslims. This is the same net result in Byzantium - the smaller powers become easy prey for the larger powers. And it was the crusading which actually helped to unify the Muslim/East against the Christian/West.
I digress. You are wanting to
avoid this internal strife and victimization of Catholic attacks, yes? You want to Byzantines to be able to hold their own at least in the Balkans and Anatolia, yes? Venice will be an issue, and Venice is the traditional ally (and sometimes enemy) of the Papacy within Italy. So there's gonna be trouble regardless. Trade, my friend, it's all about trade.
Keeping some sort of Byzantine/Roman unity is key. I figure the best possible way to achieve this is to reconcile with the Papacy. Then, perhaps the Latin Empire can be avoided. Or at least delayed. This could potentially mean that the Byzantine Emporer acknowledges subserviance to the Pope as Vicar of Christ. Not sure how that will go over with the Orthodox masses...
Anywho, assuming it IS done somehow, we can then move on to answer your question of
how it will be perceived by the rest.
Turks = Bad, dang, dirty barbarians/Muslims
vs
Byzantines = annoying, scrappy Roman Christians.
I figure the Byzantines will still want to keep all/most of their lands in the Balkans (which corresponds pretty well with what the Ottomans were able to conquer); they will likely NOT want anything in North Africa, however. Africa by this point is pretty outright against non-Muslims. Iberia was getting its head kicked by a powerful Muslim state, Egypt was a pretty scary enemy of the Crusades, and the in-between is just useless, really.
Jumping back into Italy will only bring about more pains-in-the-ass for any Byzantine emperor. Their presence will spook the Pope, who will incite more anti-Orthodox nonsense, and that whole mess of Italy will kick up a fuss. Nevermind that at this time the Holy Roman Emperor will likely come howling too, at least in part.
The Hungarians might be an issue, though. Within a century, however, the Mongols will come screaming into the scene and change up a whole lot. So long as the Byzantines play their cards right - maybe the "Shield of Christendom" and garner more support. But again, a lot might depend on the Schism.
By 1270-ish, the Mongols are wrecking shit in the Mid-East and Anatolia, as well as the otherwise Orthodox/Pagan bits of Russia-Poland-Romania area. This kind of massive movement is gonna cause ripples across the board and the Byzantine "Shield" will likely not be able to hold it - much like OTL.
Hmm... so I've talked myself back in a circle. I don't see how they can do what you're looking for and survive long enough to become more than a footnote, failling power over the next couple of centuries. A little longer, maybe, a little stronger in spots, but for the most part it will fall under the yolk of some larger power eventually. Just depends on whom.
