My starting point here is that Antony's Civil War (aka the last war of the Roman Republic) ends in either a stalemate or a partial victory for Antony and Cleopatra. Unless I'm missing something, I'm doubtful that they could have gone all the way to Rome. But perhaps they could have forced a settlement that saw Egypt regaining its independence and perhaps claiming most of Rome's eastern territories.
The most important question, aside from how this could have happened (I'm not an expert on the battles in question) is how much territory does Egypt gain. The Levant and Cyrenacia are the easiest, although I doubt Rome would tolerate losing Tunisia. The big question is how much of Anatolia goes to Egypt.
One definite possibility is that this defeat proves fatal to Octavian. This might result in Lepidus displacing him, although I don't know if Lepidus could have made himself Emperor. We might see the Republic limp along for some time, or even undergo different reforms, becoming a dictatorship but not an Empire. It's also entirely possible that the Republic would completely fall apart, with independent states appearing in Gaul, Iberia, North Africa, and Greece.
In the short run, Egypt could eclipse Rome as the dominant power in the Mediterranean. Rome was dependent on Egyptian wheat, and that would give Egypt immense leverage, possibly enough to ensure that Rome would not try to take back its lost territories. However, Egypt would now face pressure from Rome's perennial adversary: Parthia. I think that sooner or later, Parthia would manage a significant win against Egypt. It's also likely that Armenia would reassert its independence, and maybe even expand into Anatolia.
If the Roman Republic (or Empire) survived, it would probably push harder to expand elsewhere. More effort would be expended against Germania, although I'm doubtful that there would be a different outcome. Britain would either be ignored, or become another focus for expansion. Rome might also try to make more use of what it still had. In particular, to alleviate its dependence on Egyptian wheat, Rome would probably make more of an effort to settle Gaul and improve its agricultural output, reducing their dependence on Egypt. This alternate Roman state might not be as centered on Rome, with other cities in Italia, Gaul, and Iberia becoming more prominent. Perhaps they eventually settle on some kind of federal system, becoming a Roman Federation. Of course, this could also give those regions the ability to break away. Finally, a Rome that was limited in eastward expansion might try an expedition into the Atlantic. Discovering America is almost certainly out of the question, but maybe Rome could hug the coast of Africa and make contact with sub-Saharan Africa?
Egypt would probably lose its independence eventually, but it might be able to outlast Rome. The key is Anatolia. An Egyptian Empire with a firm grip on Anatolia (at least part of it) could probably withstand both Parthia and Rome for a long time. Of course, this might also produce a Roman-Parthian alliance against Egypt, which almost certainly would lead to Egypt losing. When Rome eventually fell, we almost certainly wouldn't see a Byzantine successor state in Greece - although, come to think of it, maybe one emerges in Iberia and Gaul?
The most important question, aside from how this could have happened (I'm not an expert on the battles in question) is how much territory does Egypt gain. The Levant and Cyrenacia are the easiest, although I doubt Rome would tolerate losing Tunisia. The big question is how much of Anatolia goes to Egypt.
One definite possibility is that this defeat proves fatal to Octavian. This might result in Lepidus displacing him, although I don't know if Lepidus could have made himself Emperor. We might see the Republic limp along for some time, or even undergo different reforms, becoming a dictatorship but not an Empire. It's also entirely possible that the Republic would completely fall apart, with independent states appearing in Gaul, Iberia, North Africa, and Greece.
In the short run, Egypt could eclipse Rome as the dominant power in the Mediterranean. Rome was dependent on Egyptian wheat, and that would give Egypt immense leverage, possibly enough to ensure that Rome would not try to take back its lost territories. However, Egypt would now face pressure from Rome's perennial adversary: Parthia. I think that sooner or later, Parthia would manage a significant win against Egypt. It's also likely that Armenia would reassert its independence, and maybe even expand into Anatolia.
If the Roman Republic (or Empire) survived, it would probably push harder to expand elsewhere. More effort would be expended against Germania, although I'm doubtful that there would be a different outcome. Britain would either be ignored, or become another focus for expansion. Rome might also try to make more use of what it still had. In particular, to alleviate its dependence on Egyptian wheat, Rome would probably make more of an effort to settle Gaul and improve its agricultural output, reducing their dependence on Egypt. This alternate Roman state might not be as centered on Rome, with other cities in Italia, Gaul, and Iberia becoming more prominent. Perhaps they eventually settle on some kind of federal system, becoming a Roman Federation. Of course, this could also give those regions the ability to break away. Finally, a Rome that was limited in eastward expansion might try an expedition into the Atlantic. Discovering America is almost certainly out of the question, but maybe Rome could hug the coast of Africa and make contact with sub-Saharan Africa?
Egypt would probably lose its independence eventually, but it might be able to outlast Rome. The key is Anatolia. An Egyptian Empire with a firm grip on Anatolia (at least part of it) could probably withstand both Parthia and Rome for a long time. Of course, this might also produce a Roman-Parthian alliance against Egypt, which almost certainly would lead to Egypt losing. When Rome eventually fell, we almost certainly wouldn't see a Byzantine successor state in Greece - although, come to think of it, maybe one emerges in Iberia and Gaul?