In a Web-miniseries I want to create, Les Superbes Germains Lacasse (I want that the original title to be in French) (the title means The Super Lacasse Siblings), there is a strange country that can be called a quadrumvirate, and that is both a diarchy and a double republic: there are four Heads of State that share the same power: two monarchs (one empress, and one male emperor), and two presidents (one female, and one male) (when a non-binary person and/or a genderfluid person wants to become Head of State, there are three or four monarchs, and three or four presidents). Each Head of State has one Head of Government (these people are called the Four Prime Ministers).
Also, in this country, the monarchs are not referred as "Your Majesty" or "Your Highness", instead, they are referred as "Mr. King" and "Ms. Queen". These monarchs are both elected and constitutional.
Finally, not only citizens are supposed to do the reverence to the monarchs, but they must also do the reverence to the presidents. Nevertheless, (I want the original language of my work to be French, my native language) citizens tend to use the French pronoun tu (homologous to the archaic English pronoun "thou") when addressing to any of their Head of State.
So, I wonder why would a country have both a monarch, and a president (or, alternatively, both multiple monarchs, and multiple presidents).