The unfortunate side effect is that the world would probably die with it.
Hernán Cortés declared himself victorious over the Aztec Empire on August 13, 1521, when he and Xicotencatl the Younger took Tenochtitlan.
Had he known how wrong he was, he likely would have left the Aztecs to their bloody work.
The few scholars in the know differ about what happened. Most agree that the Aztecs encountered ancient deities and accidentally awoke them, but managed to find a way to placate them with blood sacrifice. Some theories claim that the conflict between these deities shaped the New World — as the Mexicas saw a vision of an eagle eating a snake atop a cactus, a symbolic battle for the identity of the young United States centered around whether a serpent or eagle should be the national animal. One theory theory tells a convoluted tale of cocaine mummies and a pre-Colombian trade route between Egypt and Central America, claiming that the mummified corpses of the deities Horus and Set were conveyed to the New World, and their spirits forced nightly to do battle with Apophis. (Some wags, reading too many Conan the Barbarian tales, claim Set is the snake-deity.)
Whatever the case, the original Mexica people awoke something great and terrible, something which had to be placated with bloody sacrifices. When Cortés destroyed the Aztec Empire, he destroyed the last members of a cult dedicated to keeping these deities asleep (or providing the energy they need to keep the world turning, depending upon the source).
Having uncovered the secret, Cortés and his men were forced to take up the duties of the priesthood. What had begun as a tale of God, gold, and glory ended as a horror story. Fearing for his soul, he contacted the Catholic Church.
Based on this information, a secret cult arose within the Church. Dedicated to keeping the dread gods asleep and keeping their existence secret, this group of monks started the terrible work of providing new sacrifices. The cult likely took the name of the Divine Order of the Mediatrix in the 17th century, attempting to forge a symbolic connection with the grace of the Virgin Mary as well as the comparatively recent (and culturally native) Our Lady of Guadalupe and the pagan goddess Tonantzin that preceded her.
The Divine Order of the Mediatrix is a small order of monks with a special dispensation from the Holy See to perform the sacrifices necessary to keep the world turning, and to hide all evidence of this conspiracy. Traditionally, the Pope is unaware of the Order of the Mediatrix, although some certainly are. Typically, it is the quiet ranks of scholars, cardinals, and administrative personnel that organize and maintain the secret.
As for the order's methods, they are just as bloody as one might imagine. Numbers vary, but one might estimate roughly 40 serial killers active at any given time worldwide. At least one is probably a Mediatrician. It is also likely that a sizable portion of the missing persons reported each year (roughly a million, according to some reports) are similarly victims of the Order of the Mediatrix.
As for using this in a game, an early modern game might feature the Divine Order of the Mediatrix as it struggles to survive and perform its grim work. A modern-day game might feature the Mediatricians as the current conspiracy of killers for Christ, struggling to maintain their faith amidst modern technology and raw, cosmic horror.
In addition to the order of monks from which the Divine Order of the Mediatrix derives its name, the Mediatricians are aided by a cadre of exorcist-priests, therapists, and lay people who are necessary to keep the organization running.
(You can probably blame this all on my recent viewing of The Cabin in the Woods, along with my recent reading of The God That Crawls. I'm going to say that there's probably a fair share of Delta Green, too.)