Coalition Governments in the US

Here is a prediction. In the next few years, both the Republican and Democratic parties will split and the US will have four political parties. While split control of the presidency and the legislative branch has created a sneak preview of what coalition government would look like, a true coalition government where there the traditionally super strong executive power of the President will be shared across various parties/factions will be extremely strange to see. 

Of course, this has been the default in Germany and much of Scandinavia for ages.

The Trump Party - Conspiracy theorists primarily (Trump / Cruz / Hawley / most of the Republican House)

Republican Classic - (Sasse / Cotton / Collins / Kemp)

Democratic Classic - (Biden / Harris / Pete / Klobuchar / Booker)

Socialist - (Bernie / Warren / Ocasio-Cortez)

My read is that if the election was held today, the vote split would be: 35% / 13% / 32% /  20%.  

Would Rep Classic and Democratic Classic be able to come together to form a joint government?