In Finland^[1] we have a semi-presidential system, somewhat akin to France, and the power of the presidency is a constant source of debate. Its power has been significantly reduced in recent years, but there are still frequent clashes of jurisdiction between the parliamentary government and the president, particularly in foreign policy, which is the primary field of concern for the president.
(Of course, Finland has proportional representation, with no party even daring to dream winning 30% of the seats, so coalition governments and consensus are a must.)

^[1] Yes, I have an amazing ability to reduce absolutely everything to Finnish analogies.

"The basis of optimism is sheer terror" - Oscar Wilde