Religious orders might be an unconventional option for a MoSh campaign either as players, allies, or adversaries.
EYES OF UTU (EVANGELICAL SOLARIAN CHURCH)
The Solarian Church is a very big tent; while major sects usually believe that the archetypal sun god ties humanity to the world. Some sects believe in a literal sun god, some venerate many ancient terrestrial sun gods as manifestations of Sol Invictus, and others focus on the metaphor. Also, some deal drugs and work with organized criminal groups. The Solarian church is not inclined towards secret orders. It is generally well regarded by ordinary people and its doctrine doesn't threaten most established power structures it also has no central authority beyond its local leaders.
Nevertheless, some Solarians form secret groups to back up the law-giving aspect of the Unconquered Sun. The Eyes are representative of one such group. They are concerned with matters of divine justice (read: vigilantism) but also take the lifegiving aspect of the Unconquered Sun seriously. They assist spacefaring communities when life support is failing and feed the hungry when the Corps would leave them to starve.
ORDER OF ST. LAWRENCE (NEOCATHOLIC CHURCH)
The Neocatholic Church is focused on preserving human heritage and venerating exemplars of human virtue (the Eastern Orthodox Church has absorbed those Roman Catholics who follow more traditional doctrines). Being an entirely secular religion, the NCT doesn't have many secretive groups but it does have the enthusiastic support of many of the Exile States.
The Order of St. Lawrence is responsible for the NCT's efforts to secure and recover what human heritage that survived the mistake. The group's public face includes culinary historians, cryptographers, dead language specialists, archivists, and programmer/priests (and priestesses) familiar with ancient digital file formats. The Order's less public face includes archaeologists, art thieves, and IP pirates who seek to pry back humanity's heritage from the corps by any means necessary. The secret wing of the Order is very secretive and very well-armed. They are also in possession of a NEMO device.