In several states with such a semi-presidential system, if the prime minister and the president are oppositional, the prime minister will take primary responsib

In several states with such a semi-presidential system, if the prime minister and the president are oppositional, the prime minister will take primary responsibility for more domestic matters like education and health, whereas the president will take primary responsibility for most national/international matters like trade, the economy and foreign affairs. When the president and the prime minister are allied, the president often takes the lead on most matters. I have never seen a semi-presidential federation, and my intuition has become that in a semi-presidential system, the president is holds the responsibilities of a prime minister/president of a federal government, whereas the prime minister is holds the responsibilities normally given to the premier/governor of state - as if there's only a single state and a single legislature for the state and the federation. Of course, as with federations, the exact boundaries will depend on the country concerned. Whether Haiti works that way, I cannot say.