The Belgian constitution builds the monarchy on a single principle: the King's person is inviolable and his ministers are responsible. Nothing the monarch does has legal effect unless a minister countersigns it, which means the sovereign holds no independent power over legislation, appointments or policy. On paper the role is entirely ceremonial.
In practice there is one moment when it is not. After a federal election the King consults the party leaders and appoints an informateur to establish which coalitions are arithmetically and politically possible, and then a formateur to negotiate one and become, in most cases, prime minister. In a country where a majority requires parties from two separate electoral systems that have never faced the same voters, choosing who is asked to try is a real decision with real consequences.
The power is soft and it is bounded. The King cannot impose a coalition, and every appointment he makes is countersigned by an outgoing prime minister who remains responsible for it. What he can do is sequence the attempts, decline to end a mission that is making progress, or end one that is not. During the long formations of the last two decades those choices shaped the order in which options were tested, which in a negotiation of that length is not a small thing.
The convention that makes it work is silence. Palace consultations are not briefed out, participants are expected to keep the content private, and the monarch does not comment. That confidentiality is the whole asset: it gives party leaders somewhere to say what they cannot say publicly, which is exactly what a negotiation between blocs that cannot punish each other at the ballot box requires.
The rest of the job is the part the public sees, and it is deliberately unpolitical. State visits, ceremonial duties, the national day on 21 July, the annual Christmas address. The role has narrowed steadily across successive reigns, and it is generally accepted that this is what has allowed the institution to survive in a country that agrees on very little else.