Think about the situation a bit deeper.
Obviously the Libertarian Party wouldn't accept this if their only contribution was a "bodyguard". It would be a coalition government, similar to how other countries' Parliaments occasionally have. There would be Cabinet positions chosen by Green Party officials, and Cabinet positions chosen by Libertarian Party officials. Whether it is random or specifically decided which party chooses which position would be up to them. E.g., it could be like boys picking two baseball teams from everyone present.
For example:
Green Party get first choice, and chooses Attorney General. Libertarian Party then chooses Secretary of Defense. Greens choose State Dept and Libertarians choose Commerce Dept. Etc.
Each Party fields the candidates of their choice for the Cabinet positions they chose. If the Senate doesn't affirm one, the same party chooses another candidate.
Since that whole explanation can't fit in a signature line, I abbreviate it as follows: