Moirallegiance is a process that occurs between a troll and their "moirail" (also informally known as a palemate). It is a form of guardianship, but it isn't simply about being platonic soul bros forever: see Andrew's comment. They are a protector of their moirail's heart. They keep each other grounded.
A troll is compelled by fate to watch over their moirail and keep them in line. Despite the platonic appearance of this role (as we currently understand it) to us humans, trolls consider it a type of romance. Moirallegiance may also entail pacifying one's partner, like what Nepeta does for Equius and Karkat for Gamzee. Moirallegiance may translate (roughly) to our human concept of "love for best friend" which contains no intent for reproduction.
It is one of the two conciliatory relationships, the other being auspistice. As it is a redrom, it is linked to pity in the same way that black romance (blackrom) is linked to hate.
Moirallegiance is a portmanteau of "moira" (fated) and "allegiance." This kind of relationship may be most similar to the Greek concept of στοργή (storgē), or "like family," which can be defined as "a desire to ensure the well-being of another."