Anthony Kaldellis' new book, The Byzantine Republic: People and Power in New Rome (2015), dealt with this exact issue. Broadly speaking, he argued that the Byzantine empire was not as theocratic as some historians have described it and that there was a lot of continuity with its republican past. I absolutely agree with him on that point, since Byzantine studies is a rather conservative field and a lot of orthodox positions have yet to be challenged: the usage of 'Byzantine' is a case in point, as I think it would be more accurate to call the empire a Roman one, even if the usage of 'Byzantine' is perhaps more convenient. In his book, Kaldellis argued persuasively that the empire was a monarchical republic and that throughout the empire's existence (though he only examined the eastern Roman empire from the fifth century onwards) the emperor's popularity proved to be decisive in his elevation and/or downfall. He for instance noted that many sources treated emperors more as magistrates rather than as autocratic rulers, as well as the way legitimacy was often granted through popular acclaim.

This extract should be enough to outline his arguments:

They [the Romans] perceived the transition from the kings to the Republic and then to the monarchy in terms of regime change within a single politeia, the res publica Romana, and identified the different phases by their governing offices. Unlike modern scholarship they had no doubt that a continuous history of a single politeia extended to their own present, and they could and did trace every step in that history. They thought of the politeia as theoretically prior to the specific type of regime that ruled it at any particular time.

This I suspect can be applied to the historiography of the Principate/Dominate as well, as it would also be possible to argue in this case that our distinction between the republic and the empire obscures how much continuity there was. Again, from Kaldellis:

To quote Flower again: “The Romans themselves did not have a way of labeling their government with terms that specifically designated a republic... Moreover, res publica was the term still employed to describe the government during the subsequent ‘imperial’ period, both by emperors and by their critics." What we call “the Republic” was for them but one regime of governance in the long history of the res publica. The res publica itself was what underlay regime-change. Thus, we can draw a distinction between “republican,” which points to the original meanings of res publica, and “Republican,” which refers to the period and institutions of one phase of Roman history

This sounds like an argument based on semantics, but I think it's quite important to distinguish between our understanding of the 'Republic' and the historical concept of the republic. Well, this sounds interesting to me anyway, maybe the classicists around here have a different opinion?

This refreshing argument is however very new and there are as far as I know no reviews or responses to this work. I suspect this will get a lot of pushback from other historians, but this book rightfully raised many important questions. Too often, historians dismiss the people's actions as 'mob rule' or see the citizenry only as tools used by the elite in their political machinations, rather than treating the inhabitants of the empire as political agents in their own right, so I think Kaldellis is right to place his emphasis on how the people made their political choices rather than focusing exclusively on the elite. This is not to say that I agree with all of his premises, as I think that Kaldellis did not explain well the role of the army and the elite in influencing the people; he is correct to de-emphasise their role, but how far should we go? Did Kaldellis go too far the other way? Furthermore, popular sovereignty seems to me to be a very tricky concept to unpack and this relatively short book definitely did not cover everything, particularly on how this so-called monarchical republic evolved over time, as I doubt that republican traditions always had the same influence throughout the empire's history.

Regardless of my misgivings however, I think it is an excellent book and that it raised many important questions for future historians to answer. There is more work to be done certainly, but Kaldellis made an important contribution to our understanding of the Roman empire, especially as this topic has received barely any attention previously.