Canada has first past the post and three parties. They are basically what was outlined at the start of this chain: a so-called socially & fiscally conservative party (I say so-called because like most conservative parties in many western democracies, they happily cut taxes but not spending so they regularly run deficits, which isn't fiscally conservative); a socially progressive and fiscally conservative party; and, a socially and fiscally progressive party. The centrist party that is socially progressive and fiscally conservative is the normally governing party. The more conservative group tops them from time to time while the more progressive is the Canadian political equivalency of the Chicago Cubs.

The balance works because when the centrist doesn't have enough votes to be a majority, they generally get support from one side on fiscal issues and the other side on progressive issues. Things still pass.

The problem with America getting to three parties isn't first past the post -- the problem is that the groups that want three parties aren't the centrist. It is progressives and conservatives that want to ditch centrists from their party. The result is that both sides are in a game of chicken, knowing whichever goes first is worse off as the other side will keep their centrists and form government.

If the centrists abandoned the Republicans and Democrats, that is, they were the ones that wanted three parties, then America could transition to three parties.

That seems very unlikely, except... Who doesn't want to line up behind Trump? Conservative and moderate Republicans. Who has a large grassroots campaign trying to change the Democratic Party, including attacking the moderate Democratic nominee? Progressives.

The transition to three parties is closer than ever, because moderate Republicans are closer than ever to swearing off the racist, religious, and crazy branches. And moderate Democrats really aren't socialists and don't want socialism. When moderates form both parties want a moderate party, it will exist because they have the least to lose -- make people pick between three and the middle generally wins.