The answer of course, is simple, you build a portal to another world!
That is exactly the plot of his 2003 novel, Conquistador.
Starting in 1946 we follow a young John Wolfe VI as he is living in Oakland California, convalescing from his time serving in the Pacific Theater of WWII after getting some Japanese lead free of charge on Okinawa. Suddenly, while tinkering with a radio he is repairing, he opens a portal to another world. He steps out and finds a North America where no European has ever set foot. Thinking of this rich and unexplored world, he rounds up some buddies to explore.
Meanwhile in 2009 Oakland, Fish and Game wardens Tom Christiansen and Roy Tully are attempting to solve the mystery of how so many smugglers are getting their hands on the pelts of endangered species. Only to be drawn into a complex conspiracy involving two worlds and a pretty and (very literally) otherworldly special agent.
The story kicks off very quickly as our two hapless Game Wardens try to unravel the mystery they are presented with and young Adrienne Rolfe, the granddaughter of this modern day conquistador and a plot to control this gate from inside her home country that John Rolfe has founded, the Commonwealth of New Virginia.
I'll say this, the book hits the ground running with action, intrigue, suspense, and surprises and doesn't stop until a brief interlude in the middle. Stirling is on his game here and he runs the gambit from a crime drama procedural to describing battles with small unit tactics with varying things like P90s to flintlock muskets and bow and arrows, combining that with history and a nature documentary. It shows an impressive array of knowledge and a commendable depth of research. Frankly that level of detail, which doesn't clutter up the plot, should be applauded.
That Stirling also sets the scenes so well, describing sweeping vistas of untarnished nature and the habits of wildlife never seen in our world, is impressive. The alternate patterns of animals, the different geography, and the sweeping beauty of a world largely untouched by man is amazing and paints some stunning views of a world we don't know.
Now it is funny because it is an alternate history in a few ways. The first is how John Rolfe VI is descended from the John Rolfe of Pocahontas fame. In our timeline his lineage died out with his sons' only daughter, here the family lives on if slightly unremarkably until 1946 when Rolfe finds this new world.
This new world on the other hand is a total alternate history having changed so much from our own it is unrecognizable! In this world Alexander the Great and his Macedonian Empire live on as Alexander dies in 280 BC, rather than in 323 BC as in our own time period. In his prime he has conquered what would be the Roman Empire, turned back the nomads of the east, and settled his people as far west as Spain. His successors manage to keep the Empire running until it bursts in about 300 AD. That though changes the world as we know it, it is a Greek influenced Europe (trapped in the late medieval period) with numerous "barbarian" cultures floating around the periphery and cultures displaced the world over, with Indo-Iranian peoples in Manchuria! No Christianity, no Islam, and certainly nothing compared to the world we know. Then with trade with the East open the post-Macedonian kingdoms have no need to go gallivanting off across the Atlantic.
All this of course, so the Old World never meets the New.
In its place has popped up one of Stirling's more eccentric societies. The Commonwealth of New Virginia is founded by men who have old school ideas of society and morality, and a very settler mindset. If you've ever read the book Guns, Germs, and Steel you'll have a very good idea of why things go wrong for the Natives rather quickly and how these settlers move in to fill the void.
It is essentially and "oligarchic aristocracy" where the nobility (in the form of the Thirty Families) control most everything, they have their associates (basically their tenants) and there are freeholders and collaterals who are direct members of the family, or are directly in the thrall of the families themselves.
The families themselves are made up the original members who settled this new world, those who came first, and later those who provided significant capital or settler populations to help boost the Commonwealth. Many of these are people with good reasons not to be caught on FirstSide (or our world), fleeing Nazis, Pied Noirs, Rhodesians, Russian communists, Serbian war criminals, and Afrikaners. All people with very different world views than what would be acceptable today.
Society is almost stuck in the 1950s in culture, dress, and attitude, with a firm agrarian basis. Cheerfully irreverent towards displacing the natives and taking their land, and rather intolerant regarding outsiders.
This comes as a great shock to the characters who end up there from FirstSide who don't want to be there, making them distinctly uncomfortable. Many people who get close to the gate against their will are kidnapped and brought to New Virginia by force. Which is how they have a small population of precisely twenty-seven black people.
All of this is delicious window dressing though for an exciting story about how some people want to change a bit of this, by force. There are thrilling car chases, skirmishes, encounters with hostile Native bands in running Wild West style fights, and references to top notch old movies. It's a geeks dream book really.
Other than one or two parts where the world building takes over the story with description, it grabs you and doesn't let go, from page one to the epilogue it is mostly exploration and daring fights.
Like I said earlier, Stirling did his homework, and I really don't want to spoil all of that for you. It is sadly underrepresented in discussions regarding his work, and really should be celebrated for how unique it is. I highly recommend it to anyone who likes Stirling's work, and anyone who enjoys action, crime, adventure, and alternate history will love this. It has something everyone can enjoy from numerous genres.
Definitely worth a read!