Can a major party (Democrat or Republican)

form a coalition with another political party?

Absolutely.

 

They can do so whenever they want to.  They would do so if their voters wanted them to do so.  THEY HAVE DONE IT BEFORE, in Schenectady, NY, 1913.

 

Look at the history around George Lunn, the popular Socialist mayor of Schenectady, NY, 1911 - 1915.   Milwaukee, Wisconsin (Emil Seidel, Daniel Hoan, Frank Zeidler) and Bridgeport, Connecticut (Jasper McLevy) also had Socialist majors in the early 20th century.  Twenty-five states had socialist mayors during the twentieth century.  In Schenectady, the two major parties, Democrat and Republican, and a third minor party, Progressives, formed a "Fusion" ticket and offered one candidate to oppose George Lunn.  The Fusion opposition to Lunn won in 1913, but Lunn ran again in 1915 and won.  No one questioned the right of the three political parties to join together to elect a candidate or promote a political platform.

 

(Information from Wikipedia, Labor and the Schenectady Daily Gazette)

 

This occurred before the American capitalist nobility used propaganda to discredit all forms of social-justice ideas.  Lunn had started his political career as a fiery reformist preacher.  His personal political platform was the same social justice that has been a common interpretation of the teaching of Jesus Christ.  Also, back then, the Socialists were more wild.  They appeared to be connected with foreign communists and advocated extremist policies.  For example, on the national level in the U. S. A., the 1912 Socialist Party advocated collective ownership of large scale industries, safety inspection of all workplaces, abolition of the U. S. Senate and alteration of the Supreme Court.  Back then, the American Socialists competed for peaceful votes when the Socialists in Europe were armed revolutionaries.  Lunn was successful on a practical level while advocating some of the same political reforms as the more famous radical socialist figures of American history, such as Emma Goldman, Big Bill Haywood, and Eugene V. Debs.  Lunn was less radical, but he got elected and served in public office where the more radical, and uncompromising, agitators never got the chance to demonstrate their ability to exercise the legitimate power of public office.  Lunn is an earlier version of Vermont Senator Bernie Sanders, a responsible and successful Socialist whose first office was Mayor of Burlington, Vermont.  A revolution by voting is better than a revolution by violence.  We have to vote against the Two-Party System, an old vehicle stuck in deep shit, in order to save ourselves, our country, and the Earth itself.