Possibly start with a Roman-Persian merger.
See my answer to this question:
how can I replace the center of power in Christendom from Rome to Constantinople?1
Part of my answer of how to make an emperor of the eastern part of the Roman empire more powerful in late antiquity or the early middle ages includes six historical events and processes to change.
And number five is:
5) Some sort of permanent peace with the Persians should be made. All of the eastern provinces in Asia, and some of them in Africa and Europe, were once part of the Persian Empire for 200 years during the Achaemenid Dynasty. The Romans made most of those regions provinces or client states by about 200 to 100 BC.
So by Late Antiquity/Early Middle Ages (about AD 300 to 600) those regions would have been ruled by the Romans for about 400 to 800 years, and thus for two to four times as long as they were ruled by the Achaemenids.
So the Roman Empire could propose to make the Persian King of Kings a co emperor, perhaps in exchange for the Roman Emperor becoming a co King of Kings of the Persian Empire, and propose that the revenues of all Roman and Persian provinces which have ever been part of both the Roman and the Persian empires be divided according to how long each region was part of each of the empires.
Thus the Persian King of kings would be acknowledged as a co ruler of a large part of the eastern Roman Empire and would gain a vast amount of revenue from it, while the eastern Roman Emperor would loose a part of his revenue but gain the prospect of peace with the mighty Persian Empire.
And perhaps to make up for lost revenue the eastern Roman Emperor could use the monsoon winds that Graeco-Roman traders used to sail across the Indian Ocean to India to invade and conquer a part of India to tax.
Anyway, if some plan to make permanent peace with the Persians succeeds, it will prevent the bloody and destructive Roman-Persian wars, and the last and most terrible war of all from 602-628, which left both empires weak and vulnerable. Thus that would prevent:
6) The Arab conquests beginning about 630. If the Roman and Persian empires are both strong and powerful and on friendly terms the Arab conquests can probably never succeed and the unsuccessful invasions will probably cease soon.
I would suggest implementing the suggestion to conquer part of India by sending scouts and spies on trading ships from Eqypt to India to study the geographic and political situation, and following it up by sending a small invasion force to seize one or more islands and/or ports to use as bases to launch invasions of various Indian countries.
Possibly the Romans could conquer and rule the parts of India closest to the Persian borders and then conquer parts of India farther from Persia and give them to the Persian King of Kings, who would only be able to administer those lands with the cooperation of the Romans ruling the lands between them and the main body of the Persian Empire. The Persian King of Kings would know that if he made trouble for the Romans he would almost certainly loose his lands in India and their taxes and thus would have a strong motive to remain on good terms with the Romans.
Possibly the Romans and Persians would join with a powerful state in India to conquer the rest of India in a triple alliance.
If the Romans and Persians peacefully rule a large part of India, then they can send explorers by sea to southeast Asia and eventually to China, and also overland across central Asia and eventually to China. So they can open up diplomatic relationships with China and eventually perhaps negotiate a union between the Chinese empire and the Roman-Persian Empires.
Or, in a period when China is divided and torn by wars between different states, Rome and Persia might conquer China part by part, perhaps in alliance with one of the rival Chinese states.
Some people think that since the Persians "hate Rome's guts" that could never happen.
So possibly the Persians did "hate Rome's guts" for centuries, just as the Samnites, Etruscans, Carthaginians, Spaniards, Gauls, Britons, Dalmatians, Greeks, Syrians, Egyptians, etc., etc., might have "hated Rome's guts" for decades or centuries before eventually becoming Romans themselves.
But it would still be possible for old hatred to turn into friendship. In the Star Trek episode "Errand of Mercy":
KIRK: Well, no one wants war. But there are proper channels. People have a right to handle their own affairs. Eventually, we will have
AYELBORNE: Oh, eventually you will have peace, but only after millions of people have died. It is true that in the future, you and the Klingons will become fast friends. You will work together.
The best time to try such an approach would be when the Roman Empire was strongest, before the Crises of the third Century AD, and before the relatively weak Arsacid dynasty was replace by the relatively powerful Sassanid dynasty in the Persian Empire.
So such a plan should be attempted in an alternate universe by a mighty emperor like Septimius Severus (ruled AD 193-211) or Trajan (r. AD 98-117). Or even attempted by Gaius Julius Caesar, the perpetual dictator, in 44 BC.
In 44 BC Gaius Julius Caesar was planning to invade and conquer Dacia and then invade and conquer the Parthian empire of the Arsacid dynasty. Suppose that Caesar decides he has bigger fish to fry than Dacia and decides to concentrate on conquering Iran first, and suppose that the conspirators do not assassinate Caesar in 44 BC.
If all goes well for Caesar, he might invade and conquer Persia. Or possibly defeat the Arsacid dynasty so badly that the King of Kings agrees to form a partnership and condominium over much of the middle east with the Romans.
If the Romans think that making permanent peace with Persia is impossible, they might decide to conquer India to gain its wealth to finance the conquest of Persia.
So they begin with sending scouts and spies on trading ships from Eqypt to India to study the geographic and political situation, and following it up by sending a small invasion force to seize one or more islands and/or ports to use as bases to launch invasions of various Indian countries.
The small Lakshadweep Islands off the southwest coast of India and the large southern island of Ceylon/Shri Lanka might make good bases.
Conquering India would be a major conquest project for the Roman Empire, but if the Romans figure out strategy and tactics to defeat Hindu armies and capture Indian cities they can do so and perhaps control an economy nearly as large as that of the Roman Empire.
Possibly the Romans might ally with one or more major powers in India to cooperate to conquer and jointly rule all of India. During the time of greatest Roman power the most powerful state in India and most useful as an ally and troublesome as an enemy would probably have been the Satavahana dynasty.
Another Roman strategy could be to concentrate on conquering northwest India and expanding north from there into modern Pakistan and Afghanistan and central Asia to prevent the Parthians form expanding in that direction.
And eventually the Romans would invade the Parthian empire from the east and the west hoping to crush it.
If the Romans, or Romans and Indians, peacefully rule a large part of India, then they can send explorers by sea to southeast Asia and eventually to China, and also overland across central Asia and eventually to China. So they can open up diplomatic relationships with China and eventually perhaps negotiate a union between the Chinese empire and the Roman or Roman-Indian Empires.
Or, in a period when China is divided and torn by wars between different states, Rome, or Rome and India, might conquer China part by part, perhaps in alliance with one of the rival Chinese states.
The main problem with an alliance or merger between Roman and China, beside the great distance between them, is the Chinese dynastic cycle.
When things were going well in China, they were often going very well, so that a century of China equaled millennia of Europe as far as the happiness of millions of people was concerned, but when things went bad in Chine, they often went very bad, so that the fall of major dynasties and the chaos which followed often resulted in many times as much death and destruction, horror and suffering, in a few decades or centuries as in all the thousand years of the middle ages in Europe.
Repeat, a century of good times in China equaled millennia of good times in Europe, but a century of troubles in China equaled millennia of troubles in Europe. When China was a good place to live in or to rule, it was a very good place to live in or to rule, but when China was a bad place to live in or to rule, it was a very bad place to live in or to rule, so that in good times ruling China would strengthen the Roman Empire but in bad times union with China would be a liability for the Roman Empire.
I believe that the usual explanation for the dynastic cycle in China is that when a major dynasty was new it ruled well and brought peace after chaos and people prospered and multiplied over centuries and eventually the population grew too large to be supported by the farming technology of the time and famines, epidemics, breakdowns of law and order, revolts, civil wars, etc. toppled the dynasty and after years, decades, or centuries of strife and dropping population levels, a new dynasty was established and there was peace and populations started to recover, beginning a new cycle.
If that explanation is the correct one, then it would be very difficult for the Roman Empire to maintain the stability of an allied or conquered China.