@JLeslie In Canada, the Prime Minister is the leader of the party that has the most members of parliament (MPs) elected in the general election. The PM is usually the elected MP in his own riding – if he somehow manages not to get elected in his own riding, some junior MP will give up his seat so that the PM can be elected in that riding in a by-election. No direct vote is cast for Prime Minister, the point being that we are meant to be choosing individual local representatives instead of parties – although in practice many people choose along party lines just as in America. I think that because we have multiple parties with serious support from the electorate, there is less strict partisanship than in the US; a liberal really does have 3 parties to choose from, and is not likely to feel a staunch loyalty to a specific one. Lately, there is only one conservative party, but the parties change over relatively short periods of time (i.e., shorter than the time it takes for the US parties to change their focus).
In other words, the system is designed to reduce partisanship – but if one party has a particularly impressive and charismatic leader (or a particularly odious one), then people will often ignore their local representation and vote according to who they want (or don’t want) to be Prime Minister.
Of course, technically the Prime Minister is appointed by the Governor General (the Queen’s agent in Canada), but if the GG tried to choose outside of the system that has now become a long-standing practice… we would have to have some very stern words over a cup of tea or something.
Edit: So to answer your question more directly – we don’t have proportional representation either, and minority governments are quite common here, and often welcomed. Because we have more than 2 parties, who are after all not as polarized as in the US, there is usually more cooperation when a minority government is elected, and it has the effect of softening the extremes.