The government of Arbre outside the maths.
During the time of the novel (A.R. 3689 - A.R. 3691) the political situation in the Saecular world is not described in great detail, although a number of clues are given. Arbre has a world government, a Federation of large political units. These "national" units are likely known as Tetrarchies (although there are eight of them). Also mentioned is The Plenary Council of Recovered Satrapies, which may be some type of legislative body; the Eleventh Circle of Arch-Magistrates; and Prefectures.
The world government is able to coordinate a Convox of Avout from all over the world following the discovery of the space craft in orbit around Arbre, but there remain "national" distinctions. Identity documents are required for passing from one continent to the other, although numerous smugglers exist to bypass this on the route over the north pole. There also remains a significant military presence, including the active mobile ballistic missiles which are used near the end of the novel to send the members of Cell 317 into space. These missiles are said to date from "the last major war."
Events near the end of the novel, such as the Antiswarm, the disbursement of Avout from maths into small groups (Cells) hidden in Saecular society, and also the success of Cell 317 successfully infiltrating the Daban Urnud and disabling the World Burner weapon, lead to a new balance of political power on Arbre.
The world government in place at the end of the novel Anathem is based on two Magisteria, the existing Saecular Power and the power of the Mathic System. It should be noted that during the initial construction of a mathic, cloistered set of enclosed buildings named after Saunt Orolo, some Avout prepare for a future war between the two Magisteria. However, this is appropriate for the Mathic mind, which stresses a Long View of history, and doesn't necessarily suggest a belief that a future war is imminent, or even guaranteed to happen. The avout simply prepare for every imaginable contingency, because they're building a structure intended to survive for tens of thousands of years.