I’m confused by this question.

Like having a Republican president and a Democrat Vice President or vice versa?

@SergeantQueen Yes. Also could be both equally in charge as a dual leader. A government that one picks what side they are on without voting, and having half the population upset.

A choice not just a vote.

Like a half and half pizza. Or even every
slice custom made for each person. Pizza by the slice. Where you could choose your side.

In Germany, it is quite common for the winning party to form coalitions with smaller parties, including contracts between the parties on how to vote on certain issues, to ensure that they have the majorities necessary to pass laws.

A coalition government is not uncommon in parliamentarian governments, such as Germany, as @ragingloli mentioned, or Canada, Great Britain, Japan, Latvia, and New Zealand, to name a few.

Ancient Rome had its triumvirate (= 3-man rule). Other societies have had them at various times, some even into the twentieth century.

I don’t know of any two-headed structures, though, unless you count those where the head of state and the head of government are two individuals (say, queen and prime minister). But those are not leaders of different constituencies.

Having two opposing parties occupy equal roles at the top, though? Sounds like a recipe for disaster to me. Co-chairmen with equal weight can generate unresolvable conflict even in a twelve-person committee. Somebody has to have the final say.

I never heard of any.

I was wondering why some countries have both a president and a prime minister, considering they’re both on the same side.

In the first four United States presidential elections, the second-place runner was made Vice President. The opposition at the top caused friction and the system was changed with the twelfth amendment to the Constitution. The 1804 election was the first under the new rules.