I see that the cliché solution to this problem is to have all of the PCs be Magistrates. I'd rather not go that route, though.
What are the more interesting ways to get the PCs to work together while being from different clans?
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The default is the Magistrate campaign.
This mode can be applied to non-magistrate activities, too, where a group of clans each send an Imperial officer some people to use for a joint goal. Magistrates are the most common, but expeditions into the shadowlands, the Burning Sands, or Naga turf are also good candidates. One must, however, keep in mind that such a campaign's stated goal is really not part of play; once accomplished, the group logically should be dispersed back to their home clans.
Another method, if the PC's are "mostly" one clan, is have the others be local ambassadors of the other clans, or be hostages for the ambassadores of one's own clan. More flexible, and in fact quite authentic, the hostage and ambassador modes are excellent for one or two in a larger group.
Even more afield, one can have some PC's be "hired swords" of the daimyo, rather than properly sworn. Sometimes, purchased from their own Daimyo, sometimes rōnin hired. Some can even be captives who are on parole.
What we did in my previous campaign was the following : each of the different clans our characters were from were sent to the same big event, each one for a different reason (some had goals that were the same that other characters, but didn't know about it). Then, some random event happened, forcing us to cooperate because we were the only one witnessing it, and had to do something about it. Then the events make that we have to stay together, even if our final goals aren't the same.
In summary : various endgoals for characters, depending on their clans, but some event forcing the characters to be together, and bond eventually.
It really depends on how long the campaign is going to be, and how many clans you have. For instance, if you have players from 3 clans, you can have a 3 clan alliance in some big war or event. If from all the clans, then you need to make it a truly big event (Day of Thunder, Gaijin Invasion, or some other big deal).
It is possible that the characters are in different, smaller groups and they all have the same end goals, but that doesn't leave the group much reason to stick together after those end goals have been accomplished.
A more politically themed game can also work, but in this you will end up with some PvP as the players clans are at odds with each other. Other times they may be at peace, and if you do it right you could still keep the PCs together by having them be the inter-clan liaisons so they have no reason to fight, even as their clans go at it.
(I know this thread is fairly old but just in case...)
A personal favorite of mine is the "Great Master" approach. The players are all characters invited to be part of the same dojo "class" under a widely renowned master. An example that sort of ties prior ones together is something like the Emperor's guards, who come from a variety of clans but train together for a common glory and can be lumped together thereafter.
Currently I am planning a campaign where the characters have all in some way or another impressed a Kenku who wants to find its next diligent pupil. Due to the wide variety of tests, a multitude of roles and clans are able to be present. Insight Rank isn't even a problem because the Kenku look for potential and not current experience level.
I did something bold! At first the characters in my campaign started as a disconnected bunch of samurai, who were just assigned by their lords to the same quest/duty/stuff. This took a while and the Otomo whom they helped was so impressed with the different clans' samurai working together and getting s$%§ done that he created a new arm of the imperial families, called "The Fist of Heaven", which is comprised of different clans and different imperial offices.
So we got an Emerald Magistrate, an Inquisitor, a Jade Magistrate and an officer of the Imperial Legion. Responsibility always shifted to the most capable guy. We sometimes got into trouble because of the power concentration in the party, but it was a really fun campaign. It centered around how the exploits of the Fist of Heaven became famous and different daimyo requested the aid of the Fist for different tasks. Later, a second Fist was formed, and this became a thing in our empire.
With the nature of the honour based system orders from the Emperor, Daimyo or other authority figure can be used as they have to be obeyed as can some other form of honour based binding such as a personal debt for saving someone's life. If your group can deal with this it can be a particularly rewarding situation as players have to balance their conflicting loyalties in different situations.