Around the time of Christ's birth, a small Roman flotilla makes its way to North America, with those aboard the ships barely alive at arrival and the ships damaged beyond repair. With the ongoing struggle for survival in this strange new world the most important matter, the Romans decide to build a small settlement on, let's say, the island of Manahatta. After a while, contact with the local Native American tribes leads to the Romans mixing with these locals, eventually forgetting about returning home.
With the centuries passing by and history outside of North America pretty much going the same way as it did IOTL, this small Roman settlement of mixed Latin and Native American origin continually grows, gradually bringing the whole North American continent under the control of a gigantic Roman-style republic by the time the Spaniards come to the New World about 1,500 years later.