August 8th, 1880
Chile was in the midst of the Pacific War. They already had won a considerable amount of support from the British and the Germans against Bolivia and Peru, mostly to get piece of the profit in the salt mines of Southern Peru and Southwestern Bolivia. However Chile, in late 1879, had decided to begin military mobilization against Argentina, for occupying the territories in the Patagonia claimed by Chile, thus beginning a two-front war. Nonetheless, this two-front war would be different, as the Germans and British had increased support to the Chilean military much more, yet the fronts could still be fragile. In this day, a group of six gentlemen decided to meet with the Chilean president at the time, Anibal Pinto Garmendia.
When they met the president, it was discovered these gentlemen were Paraguayan representatives, who were part of a conspiracy called the "Conjura Solano", named after the ex-president of Paraguay during the triple alliance war, Francisco Solano Lopez. These men had resentment for what was done to their nation, where Argentina, Brazil and Uruguay made their country lose. Now that Chile was at war was at war with Argentina, they had a proposal... the United Republic of Chile-Paraguay, to make both a much more efficient power in the war. After some settings being put, and some talk... The president signed the proposal.
January 19th, 1883
The War had ended much earlier than the original date. That was because together both nations became militarily powerful and held off the two-front war, being even stronger with Anglo-German support, and the aftermath of it was Chile-Paraguay being an economic and political power inside South America. Now Chile-Paraguay controlled Chile and Paraguay themselves, half of the Patagonia after negotiations with Argentina, part of Bolivia and part of Peru. The following was a considerable economic growth, although a bit changed after Chile-Paraguay decided to keep more of the Salt mines instead of giving most of it to the international powers, yet giving some of it. They entered a more golden age in the 19th century. And thus, the rest of history mostly bases itself off Chile's history, with some of it mixing along Paraguayan history [i.e. Stroessner is Pinochet's advisor].
And that is simply how it is.