Consortium imperii is a Latin term dating from the Roman Dominate, denoting the sharing of imperial authority between two or more emperors, each hence designated as consors imperii, i.e. "partner in imperium", either as formal equals or in subordination, in which case the junior was often the senior's designated heir—not necessarily the natural one—and successor. In theory, this arrangement served to sharethe burden of government and ensure smooth succession, as rivalry at these moments was a major threat to the stability of the Empire; however, in practice the net result was often more civil war. Although in political reality adoption was an alternative technique to aim for the same result in terms of succession, constitutionally, this was a horror as the republic had never been abandoned in law. Thus monarchical succession in the Principate, however realistic, was officially out of the question, and this stigma persisted in the Dominate, despite the obviously monarchical trappings adopted by the emperors in this era. 'Designation', rather than simple adoption, could at least be justified by qualitative criteria.
Roman consortia imperii
An early case of one emperor, remaining "sole sovereign" in charge but designating one junior and successor, was Marcus Aurelius, who designated Lucius Verus.
Emperor Diocletian attempted an elaborate system with four emperors, called later the Tetrarchy. Revolutionary was the notion that each was to be simultaneously in permanent charge of one quarter of the empire, not just sharing in central government. The experiment did not live up to its promise, as succession was not smoothened but contention multiplied, so the quadruple emperorship was abandoned—not the quarters, which remained as administrative and military divisions called praetorian prefecture, as did the lower level, called diocese, and the smaller size of Roman provinces.
In 395 the Roman Empire was split for good, but in two halves: Western and Eastern, each under a sovereign emperor, in charge of two praetorian prefectures, each with or without a partner in government.
Both the notion of "partnership" in the form of a senior emperor and several junior co-emperors, and Diocletian's titulature, but mainly versed in Greek, became quite common in the Eastern Roman Empire, i.e. Byzantium, which lasted a further millennium after the fall of the Western Empire.