I imagine it would be a federal republic like the U.S. but instead all over the world. There would be both a president (elected directly by the people) and a prime minister (coming from the parliament) just like in France.
A constitution with a Bill of Rights and elected leaders would be a must
Not sure this could ever work but it would be interesting to see how resources are shared amongst all and vital experiments like a global space program, healthcare and nutrition for everybody would proceed
Regional history and culture, an endless number of languages and a host of stupid behaviors (nationalism, fear, greed, prejudice, warlike tendancies...etc) would make it an impossible dream, really...
We are always going to hurt each other for territory and valuables
It seems to be in our nature to fuss and feud endlessly
We will never have a one world government because the first person to scream for a king or communism is going to be painting a target on his face...leading to more fighting
I am curious as this is not something I am familiar with. What are the powers of the president and prime minister? Is one more powerful than the other?
I'm not so sure myself. I just know that France has both offices and there is a thing called cohabitation where the two leaders can come from different parties. I decided to borrow the idea for my hypothetical Earth government.
I'd assume the president has the final say, nonetheless.