AFAIK basically all other democracies function this way - using voting districts - never using direct representation to elect representatives [1].

One similar example of this is the recent German election - I believe AFD had like ~10% of the popular vote, but only won 5% of seats in the parliament.

[1] https://en.m.wikipedia.org/wiki/Direct_representation

Germany (and NZ) use MMP - everyone gets 2 votes one to select someone for their local voting district - that provides half the seats in parliament, the other vote is for a party, these votes are used to choose the other half of the seats which, taking into consideration the parties of the local representatives who have already been elected, are used to pad out the party representatives to match the country wide party vote.

In both countries there is a threshold (5% party votes) that a party must get to get party seats ... looking online in 2013 AFD (spit) got less than 5%, below the threshold, and got no seats, in the recent election they got above the threshold and were awarded 92 seats (14%) - seems to me that the system is working as designed.

But just because they got seats it doesn't mean that the other parties are required to stoop to form a government with them.

MMP is a far better system than the FPP system used in many other places, most people end up with someone they voted for representing them, unless they are in an extreme minority (<5%), and it's pretty immune to gerrymandering. Personally I'd argue for reducing the 5% threshold to the size of one seat so even more people are represented.

In Canada the Government is occasionally elected without the majority of votes. It is also where the Prime Minister is simply the "first among the caucus" and not elected in a separate election.

These are referred to as "minority" Governments.

Any other country where the election isn't based on the nationwide popular vote.

Because it was not a popular vote. And people voting were aware it was not a popular vote.

You cannot divine now how many would-be Trump voters in blue states stayed home versus would-be Clinton voters in red states.

Plus the two million eligible voters who were denied their right to vote.

But who’s counting?

It’s more fun to argue about facebook, benghazi, email servers, value voters, likeability...