PoD: Jesus's mother Mary migrates to rural Persia as a girl. Jesus preaches, not on the banks of the Jordan, but on the banks of the Euphrates. The new religion of Christianity starts spreading in Persia; the resulting social unrest in Rome's eastern neighbors, thus making them less powerful, combined with other butterflies, means that the Roman Empire doesn't collapse as quickly in around 200 AD.
Roman boat building capabilities steadily improve, and while there is no Industrial Revolution, by 400 AD boats are built that allow travel across the Atlantic, if not easily.
By 600 AD, the Roman Empire rules Macaronesia and some Caribbean islands; though those islands are rather distant from Rome, they're well-defended. Rome is technologically stagnant by 600 AD, due to steadily increasing bureaucracy. A few theocratic Christian empires now exist, the most important being the Armenian Empire.
Christian conquests from the east, and Vandal conquests from the north, steadily erode the Roman Empire over the next few centuries. The capital of the Roman Empire is moved to Nova Carthago, Iberia, in 779 AD, at the same time as the Fall of Rome.
By 900 AD Rome has lost all of its Afroeurasian territory, apart from Iberia and Macaronesia, to various conquerors. Iberia is a safe Roman stronghold for now, due to being distant from Vandals and Christians. However, it has now united all Caribbean islands under its rule - the Roman boats were vastly superior to the native designs, and the standard Roman model of incorporating all local deities into its polytheist faith has worked well. The nature of the Empire has changed over the centuries following the changing territory, and it's now a primarily naval-based power. The city of Nova Athenae, on *Puerto Rico, is a major imperial city on par with Nova Carthago. Romans have generally been teaching the local Caribbean populace the Latin language, writing, and many aspects of civilization - their choice to conquer in this area was based on the local populations being less advance, thus easier to subdue.
Iberia finally falls to Christian invaders in 1132 AD, and the Roman capital officially moves to Nova Athenae, though it had been de facto located here for some time. The states of Macaronesia, no longer located on important Roman trade routes thus becoming isolated from most civilization, all declare independence over the next few decades.
This marks a key turning point for the Empire, as the resultant Empire, ruling only the Caribbean, is both at the weakest it's ever been, and it only controls areas on one continent again. In addition, it hasn't made much technological progress since 600 AD. Most citizens are Native American in ancestry, culturally a Roman-native hybrid. This begins the long Era of Civilization, whereby "savages" are introduced to Roman civilization. Expansion onto the *North American continent is steady, with a major city of Memphis being built on the Mississippi in 1379 AD.
Roman hegemony in this area grows to a peak in 1551. Soon, the European empires come with their fabulous technology, and soon take over the Roman heartlands in the Caribbean. Many tales have been written now about the perennial loss of the Romans' homelands.
The Romans are now a land-based empire, another seismic shift, based in Memphis. Expansion into the Plains and across the Rockies occurs within the next few centuries, but their oldest lands keep getting lost to expansionist Europeans.
This perennial "expand into new territory- lose old territory" cycle occurs because the Romans are far superior technologically to the native Americans, but far inferior to Europeans.
This cycle only finally ends when the Romans rule a cold land that nobody has interest in. As of the twentieth century, the Roman Empire controls only the city of Nova Nova Roma (OTL Anchorage), and the endless frozen wastes around it.