"Republic" is not a geographic entity. "republic" is a form of government, or do I really have to go into the history of, for example, the Roman republic?

A republic is a form of government in which the country is considered a "public matter" (Latin: res publica), not the private concern or property of the rulers, and where offices of state are subsequently directly or indirectly elected or appointed rather than inherited. Today the simplest definition is that we have a government where the head is not a monarch. So yes, the US is a republic by that definition as well as the classic definition. Accept it.

In regards to North Korea and China, they call themselves republic, but that doesn't make them one. I can call myself POTUS, but that doesn't mean I am that. North Korea is Asia's Nazi Germany, while China isn't far behind.

Besides, if we start using geographic entity as definition for "republic", then oh dear gods, some people here need some lessons in geography, especially in regards to China.