As a Dungeon Master or Party Leader, how do you handle a party that is decisively split against itself so that some players insist on doing different things thus causing friction in the game?
That all depends on the reason and the split. Usually I would let them split apart and then have things hit them so they understand the power in numbers.
I've experienced that a couple of times even in big K's game. It happens for a number of reasons. One reason could be simply role-playing your character. I like to play my character feisty so she does not tend to follow orders easily. The other is a part of the group thinks a certain action is too much risk so they break to save some of the others rather than everyone dies.
A split is bound to happen when character personalities are at odds with a chosen path. The split can be major or minor but good players usually work it out. If the party can’t come to a conclusion, you as Dungeon Master have to step in and play both chosen paths to a sensible and plot-aiding conclusion. That usually brings the group together.
As my favorite dragon always tell me Split them up and kill them all The ones in armor is the best he says Crunchy on the outside chewy on the inside *smile*
I let the characters decide what they are going to do. If they wish to go two separate direction go ahead I can still get them towards the common goal even it from two different direction or styles.
This is where the term "Leader" and "Master" come in so they aren't just figure titles. Good leadership often has compromise or middle ground with a little give and take on both sides.
It can also be difficult, for a groups leader to foster unity in a party.
Some, characters if role-played correctly, are meant to create friction, where there wasn't before.
I think one way, for the leader to get back a sense of control over the situation.
Is to remind the other characters, the importance of the current goals, and mission. Also, the consequences if the characters don't complete it.
Perhaps a timely reminder, that the hand is no more immportant than the eye, or ear etc. Put simply, the characters, need to find their synergy!
If a party splits it makes the game complicated specially for the Dungeon Master. If you ask the Dungeon Master I'm pretty sure he'll want you to go back to being one party- plus its safer that way.