The South American Naval Arms Race and Civil War
Amusingly, the most vicious naval arms race would take place not even in Europe or Asia, but rather South America. The aftermath of the Great Pacific War, which was almost entirely decided on sea, not on land, made several Latin American countries realize how important naval power was. The ongoing Confederate-Spanish War also taught them this. In addition, the peace between Peru-Bolivia (often just called the Andean Confederation) and Chile left neither side particularly happy.
In addition, the Americans were not allowed to help negotiate the territorial dispute between Peru-Bolivia and Paraguay [1] because they were quite reasonably seen as too close to Peru-Bolivia. The Confederates were eventually brought in to help negotiate, but they were rather unhelpful and their diplomats were eventually pulled back due to the outbreak of war with Spain. As a result, South American nations quickly became playing fierce power politics.
The Chileans and the Paraguayans shared an enemy in Peru-Bolivia, so military cooperation between them became quite the norm. Chile and Peru-Bolivia both rebuilt their navies with gusto, gaining overseas partners. Immediately after the end of the Great Pacific War, the Chileans ordered the construction of the Capitán Prat from French shipyards. Upon its completion in 1886, it became the most advanced pre-dreadnought battleship in the New World. The Chilean Navy became so strong, that when José Manuel Balmaceda lost the confidence of the Conservative Congress and Conservative-dominated Navy, he realized that the situation was hopeless, and instead of possibly starting a civil war, he immediately resigned.
The power of the Chilean navy terrified the Argentinians, who immediately embarked on their own massive ship-building program. Relations between the two nations had been extremely poor after the Chilean delegation stormed out of the 1881 boundary negotiations with regards to Patagonia due to (correct) allegations that the Argentinians were allowing the Americans to supply Bolivian-American forces in the Great Pacific War through Argentinian territory. The Argentinians, perhaps drawing on large amounts of immigration from Italy (which only increased due to America being seen as unfriendly to Catholics), contracted with various Italian shipyards to build their own navy, ordering the battleship Independencia.
In turn, the massive Argentinian naval build-up also inspired yet ANOTHER South American nation to enter the ludicrous South American arms race. The Brazilian Navy was seen as particularly powerful, with two powerful battleships, the Riachuelo and Aquidabã. This took place under the powerful Joaquim Marques Lisboa, Marquis of Tamandare. The Marquis of Tamandare, having been literally old enough to start his service in the Navy of Portugal-Brazil, was a well-respected leader. He was put in charge of a second naval buildup that was sparked by fears of continued American intervention in South America, especially because after the Paraguay-Brazil border compromise, it had been the policy of the Brazilian government to support Paraguay against Argentinian and Andean territorial claims. It became feared that the United States would use its navy to pressure Brazil. As a result, the Brazilians added another two top-of-the-line ships to their Navy between 1885-1889, leapfrogging the Chileans into becoming the most powerful navy in the Americas.
This naval build-up would have a remarkable influence in the Brazilian Civil War of 1890. Coffee oligarchs were enraged that the Brazilian monarchy had totally abolished slavery in 1890. It was originally to be done in 1888, but it was temporarily delayed by Brazil's hosting of the Confederate States and Spain in the Conference of Rio de Janeiro, since it might have jeopardized Confederate participation by making it seem like Brazil had an axe to grind against them. Regardless, in 1891, a military coup was launched against the Emperor of Brazil by Deodoro da Fonseca and Floriano Peixoto. Largely successful, they seized control of the capital, declaring Congress abolished, the Monarchy abolished, and a new Republic established. However, several royalists were furious, especially after the death of Emperor Pedro II as he was fleeing the nation. Conspiracy theorists quickly arose, claiming that Pedro II had been murdered by the revolutionaries.
Although the army largely supported the Federalist Revolution, it was opposed strongly by the Navy. In addition, two foreign leaders both had been carefully eyeing the Brazilian Revolution. The Brazilian Navy responded by going completely AWOL, parking outside of Rio, and starting to bomb positions of the Brazilian Army. The Rio Grande do Sul revolted again, claiming that the Federalist government sought to crush their autonomy once again. In addition, Antonion Conselheiro, a mystic prophet in Northern Brazil (in Bahia) was gathering followers, claiming (accurately) that the Republic was a conspiracy of slaver oligarchs. In many cases, his men were joined by veteran soldiers who had sailed directly from the Christian Commonwealth of North Carolina to protect them. This caused his numbers to swell, which caused local Brazilian army troops to retaliate against them. Immediately, war was to also break out in Bahia.
The Brazilian Federalists still had the loyalty of most of the army. What was to break their spirit was horrifying information. The French were on their way. Princess Isabel, now to many, Empress Isabel, was an ultramontist married to Gaston, the Count of Eu, the grand-son of King Louis Philippe of the House of Orleans. Napoleon IV, unlike his father, deeply sought the support of the Orleanists in France, under Prince Philippe, Count of Paris, who also more or less inherited the Legitimist claim to the French throne as well. Prince Philippe, a genuine democrat, had even signed up for the Union Army in the American Civil War to help defeat the slavery. Napoleon IV came to him with a simple offer. If was willing to renounce his claim to the French throne, the entire House of Orleans would be welcomed back to France, given all of their old titles (minus the actual monarchy), and the French Army be immediately deployed to help his cousin, Gaston (who he grew up with in the Clermont Mansion in Surrey, England). After all, if Napoleon III triumphed gloriously in Mexico, why couldn't Napoleon IV triumph gloriously in Brazil? Best of all, the British public was remarkably sympathetic to the Orleanists, and although the British government loathed any flexing of French power abroad, they had no standing to condemn them. They tried to get the Confederates to do their dirty work for them, but this failed, especially as the Confederates were too internally divided to do anything like this. Eventually however, they found two ways to accomplish their objectives.
Worst of all for the Federalists, that wouldn't even be the only foreign interloper. Prince Pedro Augusto, increasingly estranged from the rest of the Brazilian Imperial Family, still believed himself to be the best successor to Pedro II, especially as many members of the abolished Congress actually supported him. He also became increasingly erratic. President James Blaine had been eagerly looking for a way to flex American power abroad during his regime. Inviting Augusto Pedro from exile to Washington D.C., the United States announced the reestablishment of the Monroe Doctrine in the aftermath of the American triumph in the Spanish-Confederate War, recognizing Augusto Pedro as the legitimate Emperor of Brazil. American marines were immediately deployed down the East Coast, to fight a second war in South America. Due to close American relations with Peru-Bolivia in the wake of the Great Pacific War, this translated into those nations allowing supplies to float to the Federalists, even though the Imperial Navy had mostly blockaded the nation.
The Federalists had a few surprising friends in the war. First, the North Germans pretty much reflexively opposed anything the French did, so a North German military attachment was quickly drilling Federalist Brazilian troops in the Prussian way of warfare. Italian supplies, from the reasonably powerful Italian navy, were also flowing into Brazil. The British gave them full access through Gibraltar, which quickly brought Italy and Britain closer together in diplomacy. Finally, Great Britain remained officially neutral, but one seemingly independent nation didn't. The Dominion of Canada, increasingly divided on religious lines and in the middle of a Culture War with the Catholic Church, immediately jumped to support President Peixoto of the Federalists (Fonseca died not soon after taking power as President).
Argentina most notably enjoyed close relations with both Italy and the United States, so it took the unusual position of supporting both Peixoto and Augusto Pedro. In contrast, the British public (and its donations) were largely divided between Peixoto and Isabel. That being said, the two Royalist factions didn't actually spend that much time fighting each other - they spent most of their time fighting the Federalists, especially as Augusto Pedro's base of support quickly became the far South and Isabel's base the far North. The prosperity of Pedro II's reign was to end in the most brutal spat of violence to ever hit Brazil.
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[1] OTL, President Hayes helped resolve this dispute!