Perhaps this thread is as good as any for discussing why no imperial power prior to the Arab Caliphates themselves (and them not for long) ever dominated the Red Sea as a Mare Nostrum for any long time or to the full extent of being strong on all shorelines. Apparently controlling both shores and outlet of the Persian Gulf has been a more common thing, though not generally the case either. Both at the same time is as far as I know completely unknown to history save to the degree the Achaemenids might have been said to have indirect hegemony of the Red Sea and thorough control of the Persian Gulf. Even controlling both coastlines of both bays is not the same as having power in the Arabian peninsula interior.
Anyway starting with the ancient Egyptian dynasties I'd think the potential for a Red Sea hegemony and some motivation for it existed. Controlling the northern bay, at any rate the shore opposite the west shore of Sinai, surely the greater and richer Pharaohs could afford to build some kind of fleet of ships suitable for Red Sea if not wide Indian Ocean navigation, and descend down the coasts on both African and Arabian sides, gradually securing all the major ports all the way to the strait of Yemen. That Egyptians would not bother (being deeply ethnocentric on the Nile as the place where real normal sane people lived) may make sense, but the Persian conquest would be the first time that Egypt was under an empire of such vast scope. The Achaemenids certainly had rule over Egypt and ample deep pockets to make a project of securing the entire Sea, incorporating Yemen and whatever is on the African shore near the strait mouth and hold it for generations. Now of course this lifeline to the strait might be vulnerable to being cut by the inland tribal peoples descending on the coastal ports, especially on the Arabian side, which is how such an Empire would turn toward dominating these tribes too thus gaining vague control over the interior despite this not being any sort of regime priority for any primary reason. Alexander of course was a briefly burning candle but the Ptolemy dynasty lasted many centuries and I'd think they'd get around to it by and by. The Indian Ocean trade is just such a major revenue stream; maybe not as much as I think but the task of holding the shore port towns even against hostile people in the interior is not so huge either, as long as richer Yemen is under cost-effective rule. So why not monopolize it?
The Romans I believe did try it briefly, based on my seeing maps referring to "Arabia Felix" (Yemen) and the impression this at least was briefly an exarchate province, but it did not last long despite the Empire's very deep pockets. This brings us to Islamic times of course when Arabia vaults from being a peripheral zone along a trade route to the cultural center of the world of a major fraction of humanity. The early Caliphates I suppose did claim control of both shores and the mouth, but that did not last too long either--by the time of the Crusades the Shi'ite Caliphate controlling Egypt was at odds with other powers, Seljuk Turks in Palestine and Syria and I believe independent tribes on the Arabian shore at any rate; the Egyptian dynasty definitely did not control the Hejaz coast. Nor am I sure their control extended from the Nile all the way to the Red Sea coast of Africa even.
So why is this? Is it just that the shorelands between Sinai and the southern straits are so desolate it just isn't sustainable to try to guarantee pirate-free passage, so the cost of completing the claim is too high for the limited benefit, or what?
