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The Imperial Inca Republic was a short-lived nation in South America, existing years 2023-28, with its capitol in Lima.

History:

In the late 2010's, the indigenous people of South America started revolting. In Peru and Bolivia, the revolts ended with the creation of Amerindian nationalist governments, who started persecuting non-Amerindians, driving them into neighboring nations or into special reservations. In Chile and Argentina, an organization called the Southern Inca Sovereignty Front gained control over parts of the northern regions and declared independence as the Southern Inca Republic. In Paraguay, the conflict led to the division of the country into the Guarani Republic (east) and the State of East Gran Chaco (west). In Brazil, the Amerindians were given more rights and protection by the state, but the State of Acre declared independence, something that was not recognized by the Brazilian government.

In 2023, Peru, Bolivia and Southern Inca merged to form the semi-federal Imperial Inca Republic (IIR), re-installing the monarchy and choosing the leader of the original Peruvian revolt as the emperor. The emperor, however, did not have total power, and a new one was to be elected by the people every 8 years. In 2024, the IIR annexed eastern Ecuador after pro-Inca rebel groups had overtaken the area, and this same year the State of East Gran Chaco and the State of Acre also joined.

In 2028, as a result of months of rising tensions, the empire was invaded by a coalition including the United States, Mexico, Gran Colombia, Brazil, Argentina and Chile. A more democratic and less militaristic, nationalistic government was inserted, but non-Amerindians continued to live under somewhat bad conditions for several years.

Those are more common as you go back in time although usually it is the nobility or a small subset of them who gets to vote. For example the head of state/chief of state of the UAE is one of the emirs who is elected for life by a vote of the seven emirs if I remember correctly. In the past Poland and Transylvania did similar things.

The word 'republic' is the opposite of the word 'monarchy'. You can't have both.

Any government is a republic if the leaders are not given a title of nobility. Canada is a monarchy, despite the presence of a constitution, and elected parliament, and a non-monarchic head of government. North Korea is a republic, despite the presence of a hereditary absolute ruler.

Republic has nothing to do with any other governmental structure, it is not about how things work or with whom the power truly lies, or even with how the leader gets his position. If the American President was given the title 'Emperor' while in office and nothing else changed, the USA would cease to be a Republic.