Diplomatic alignments can sometimes be very straightforward, ever since the British decided to support Chile, Bolivia and Peru had no choice but to seek support from France and Austria.
International trade is a minor issue; with such small national scales in South America, if it weren't for their rich resources and the promotion of currency hegemony, the great powers would not have taken notice of them.
One can tell by the population figures, approximately 2.2 million in Chile, about 2.75 million in Peru, and around 1.3 million in Bolivia.
With such small populations, and all being agricultural countries, expecting a large internal market is clearly implausible.
Chile is relatively wealthy, mainly due to its saltpeter exports, and its military is the strongest among the three nations.
Peru has silver, but unfortunately, the price of silver has been declining over the years, drying up the Peruvian government's purse.