It may not be entirely truthful to call the Empire—the country with which the Shine Cycle is principally concerned—or any of the individual countries of which it is composed a “republic,” as they’re elective monarchies. But other than that, and other than the districts whose governors or representatives have had their posts made permanent to give them seats in the House of Peers, each district, each country, and the Empire as a whole is arguably a republic. Each is governed by an elected legislative body (the Parliament over the whole Empire) and an elected (if in many cases life-term) executive.
What’s more, the Empire is a federal republic. Each district sets the rules for how its legislature and its representatives on the next higher level are selected, but aside from the lowest level (where direct election is nearly universal), most districts have their constituent districts’ legislative councils elect at least some of their representatives. Metaphysically verified oaths (and a strong culture of honor) keep this structure from becoming as corrupt as the somewhat similar system in medieval Venice. The one exception to this pattern is that the lowest level of Parliamentary representatives (you may recall that the lowest few levels of districts aren’t directly represented in the Imperial Parliament, to reduce its size) tends to be directly elected.
I mentioned earlier that “direct election” is nearly universal in the lowest level of districts. But the franchise—the right to vote in these elections—is not universal. Each district has the power to determine its electorate, within reason; some require only a certain age, or a competency test (similar in concept to the exam candidates for naturalization are required to take in the United States)—with no “grandfather clause”!—but far more limit it to landowners, heads of household, or those who can prove themselves financially independent (of their parents, any employers, and the government), so as to prevent any one voter from influencing another through any means more severe than persuasive argument.