The French Empire of Louisiana
Napoleon obtained the Louisiana territory from Spain in the 1800 Treaty of Ildefonso. The intention of the territory was to serve as a strategic holding to assist in the development of the Haitian Colony, in the hopes that Haiti would become a profitable source of French income. In OTL, the Haitian Revolution earns the Haitians independence; and as such, Louisiana was more cost than it was worth. Napoleon sold the territory to the United States under President Jefferson in 1803.
In this world, the Haitian Revolution is crushed by the French Military. So, Napoleon never sells Louisiana. The war in Europe continues for the most part unaltered, and Napoleon's forces were defeated in 1814. However, rather than accept surrender at the hands of his enemies. On the eve of the Battle of Paris, Napoleon fled the continent and sailed west with his supporters.
Napoleon first settled in Haiti, and continued to rule as French Emperor, though in exile. The legitimate French Government began plans to reconquer the New World Territories, but Napoleon struck a deal with the other colonial powers. He would make no attempt at claiming, attacking or conquering any other colonial possessions, as long as they kept France off his back. With their colonial holdings already unstable (With several claims of independence), a deal was struck.
All was well for Napoleon until the Second Haitian Revolution began a decade later. The Napoleonic forces eventually had to abandon the region, and fled to New Orleans.
Here, Napoleon established a base of power and loyal supporters throughout the territory. He would continue to rein as self titled French Emperor until his death in 1830. His son, Napoleon II, would rename the territory and the monarchy, having accepted that Napoleonic control over France Proper was no longer possible. Thus, the French Empire of Louisiana was born. The United States did not enjoy having a monarchy right next door, but with the port of New Orleans open to all trade, including American, the new found Empire was tolerated.
In the 1890s, a Revolution would attempt to overthrow Napoleon IV, but a compromise was reached and a Parliament Assembly was established. Today Louisiana is a moderately powerful state, with it's exports earning it the title "Bread basket of the world". Though the culture is definitely French, the years an entirely unique Louisianian culture has developed. Napoleon VI reigns on the throne, while the Conservateur party holds a majority government, with a Labor-Social Démocrate coalition opposition.