That was the case in the UK from 2010-2015. It's called a minority government. Various factions horse trade until some coalition of parties has a majority, and

That was the case in the UK from 2010-2015. It's called a minority government. Various factions horse trade until some coalition of parties has a majority, and then they rule as coalition. Members of both the Conservatives and Liberal-Democrats had cabinet seats. The main downside to this sort of process is that sometimes negotiations take a long time. Belgium spent nearly two years bickering over forming their coalition in 2010.