You have a republic rather than a monarchy. In order to establish its leadership, a republic can use representative democracy, direct democracy, fascism, dictatorship, whatever -- as long as it defines the power of the state as flowing from the entirety of its people, rather than from a subset of the population or by a divinity, it's a republic, end of. "Republic" is not a synonym of "representative democracy".
I wish I had a penny for every time I had to dispel this awful misconception in constitutional studies that only Americans seem to have.