Y'all elect a President.
In bambuland we elect a Party, the leader of which becomes Prime Minister if they win an election.
At any time, reps of the governing party can have a 'spill' and declare the position of Prime Minister vacant and have a Party-room vote.
This is what happened today.
Prime Minister Turnbull was told he no longer had the support of the majority of the Party reps, so he was gone from Prime Minister, and there will be a vote in the Party room tomorrow to elect a new leader of the governing Liberal Party [Conservatives] and that person will be the new Prime Minister.
Several Republican Party reps could walk around to President Trump's office, tell him he no longer enjoyed the support of the majority of reps in the Party and he would no longer be President, and the reps would have a vote and elect a new leader/President.
It might seem like an unstable system, but it's not, in reality.