Which counries would be classified as neither republics nor monarchies?

[2952-0-0]

The UAE is a federation of seven emirates, which are by themselves monarchies. The president of the UAE is by convention emir of Abu Dhabi, and the prime minister is by convention the emir of Dubai.

Andorra has two co-heads of state: the Roman Catholic bishop of Urgell and the President of France. One is appointed by another head of state, while the other is elected by the people of a foreign country.

[2952-0-0]

Also add the Vatican, which is somewhat unique

Vatican is without question a monarchy. Its head of state makes no pretense of deriving his (and always will be his) authority from the people. He is elected by a group of men who pray to the Holy Spirit to determine their best choice. And the cardinals are appointed by...previous popes.

Japan's Emperor explicitly has absolutely zero powers according to the post-WW2 constitution. He is purely a symbol of the people's unity and isn't mentioned as head of state. Any reference to government powers being exercised in his name is absolutely taboo.