Yeah, ALF is really abusing the word "Republic," which simply means "a country that isn't a monarchy" (thus North Korea, the US, and, Switzerland are all Republics, but the UK, Saudi Arabia, and Sweden all aren't). Republic has nothing to do with how the government is actually selected, which, in the US, is done democratically. The "Republic, not a Democracy" thing is the dumbest talking point.
North Korea is not a republic. It has a hereditary head of state.
Can you give me an example of a "democracy"? Are there any countries in the world where elected representative don't govern as representatives of the people? The referendum system would be the closest to "democracy" as you define it, no?
No, I don't think there are any countries where elected officials don't govern as representatives of the people. The referendum system would be the closest thing to a true democracy, yes.