The recent talk about a hypothetical Canada - Scotland political union has me wondering whether an old world country has ever been ruled or governed from the new world. Outside of marginal cases such as the American occupation zone of Germany, the closest analog seems to me to be my (dimly recalled and possibly mistaken impression) that at some point during the Brazilian monarchy, Portugal was run from Brazil. Is this correct?
You are correct. But as you might guess the situation was far more complicated than that.
In 1803 the UK and France went back to war. Portugal was a long standing ally of England, but wanted no part of war with Napoleon. So the regent João (later João VI) tried to walk a line of neutrality between them. João was in a precarious position as he was ruling on behalf of his insane mother Maria I, and not by his own authority.
In 1807 Napoleon demanded that Portugual join the continental system, close their ports to the British, and seize British property. João not wanting to piss off his British allies only did part of what Napoleon wanted, stoping short of seizing property. He hoped that would be enough to appease Napoleon while stil not upsetting the British too much. It wasn't enough and by November Napoleon had troops marching on Lisbon.
On November 29th, some 15,000 Portuguese royalty, nobility, and attendants boarded Portuguese and British ships and sailed for Brazil. On December 1st general Jean-Andoche Junot entered Portugal. Portugal was divided into three zones of occupation and the Spanish and French took over rule of the country.
The Portuguese populous didn't take too kindly to the occupation. In 1808 the locals joined in popular uprisings in both Spain and Portugal kicking off the Peninsular war. In 1809 the future Duke of Wellington arrived and drove the French out of Portugal. The French were back in country shortly after, and especially the north of Portugal was invaded and liberated several times. In 1811 Wellington was able to push out of Portugal for good, and the rest of the Peninsular war was fought exclusively in Spain. During these first few years it's hard to say anyone was ruling Portugal, let alone the Portuguese court in distant Rio. Rule either fell to whatever local politicians could hold, or to whichever military officer happened to be closest.
Before long the British officer, William Beresford was made Marshal and commander in chief of the Portuguese army, in March of 1809. He also acted more or less as the military governor of Portugal throughout the rest of the Napoleonic wars. As time passed this role was made more and more official, but it was largely just the Brazilian court recognizing the power he had already just simply seized. As the years passed, even after the war ended, he made trips every couple of years to Rio where he was consistently invested with more and more power.
The Portuguese were very unhappy with this turn of events. Once the war was over they had expected to see the court return to Lisbon. They wanted a return to preeminent status in the empire, and they wanted their British overlord to go away. So in 1820 when Beresford returned from a visit to the Court in Rio, he found a revolution in progress, and the locals didn't even let him off his ship. The revolutionaries demanded a return of the crown, that Brazil be downgraded back to a colony, and a constitution. By 1822 got everything they wanted.
So while nominally Portugal was ruled by Brazil for 15 years or so, in practice war and insurrection greatly limited the court's practical control. Functionally Portugal was ruled by no one or everyone depending on local conditions. Little thought was given to the opinions of the far away court in Rio. Most of all it was ruled by a British general who's authority was half appointed and half seized. It's hard to say that there was ever a point that Portugal was essentially ruled from Brazil.