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Imagine a Britain where an elected President is head of state he/she has to choose a cabinet and prime minister based on election results.
This gives rise to some improbable combinations....
1964 President Alec Douglas-Home Conservative
Prime Minister Hugh Gaitskell Labour
Home as an hereditary peer who gave up his seat in the Lords to become President is unhappy with giving the 'socialists' power but the narrow election win for Gaitskell means he has to share power.