Well part of the problem is that there isn't total agreement on what a republic is. By some definitions it's basically anything that isn't a monarchy. Some medieval republics didn't have elections and instead chose their officials by sortition, which is essentially a lottery. China and North Korea do have elections, but they're total shams (and North Korea is basically a monarchy is a thin coat of republican paint, since by law they can't have any leader that isn't descended from Kim Il Sung).