THE HOLY GRAIL WAR
The teams must get involved in the Fourth Holy Grail War and win it.
What is the Holy Grail War? In a nutshell, a complex magic ritual in which seven wizards summon each one Servant, a heroic figure of history or myth magically brought to life and given random-ass superpowers. The ritual requires that the Servants fight each other to the "death", since the destruction of the Servants releases magic that is used to form the "Holy Grail", a wish granting device that is supposed to grant a wish to the winner of the conflict. This can be accomplished through direct battle or by killing the summoning wizard, since this deprives the Servant of his power source generally causing it to vanish away.
What do the teams know? They know that the conflict is secretly fought within the boundaries of the city of Fuyuki, Japan, under the supervision of a Catholic priest who makes sure things don't get too out of hand. They know the bare generalities of the Holy Grail War (such as the Servant classes) and magic in the setting (including abilities that any magic user in the setting can be reasonably expected to know, such as hypnosis, use of familiars for long range scouting and body enhancement). They do NOT know the identities of the summoned Servants, nor do they know the identities of the summoners.
What do the teams get to begin with? 1) Standard equipment; 2) Characters in the Mage slot get to learn a spell that lets them become Masters of a Servant, under certain circumstances (see below for details); 3) Nothing else, not even the ability to speak Japanese, so if there are no Japanese speakers in a team, they might as well start buying Japanese-English dictionaries.
What must the teams do?
a) They must identify at least one of the summoners. Seeing that the Masters mostly try to stay in hiding for security reasons, this would require both detective work and probably some hocus pocus to find mystic trails.
b) To win the Holy Grail War, a wizard HAS to have a Servant, but by the time the teams show up all have been summoned. So they HAVE to steal somebody else's Servant, either by intimidating the Master into surrendering control (which is a fairly tricky thing to try) or by taking somebody's Command Seals, which are magic symbols gained by the summoners and which give them a link with their summoned Servant and a measure of control.
c) After getting a Servant of their own, the team has to try to defeat the other Masters and Servants, to make possible the formation of the Holy Grail, while also being discreet enough to keep the existence of the supernatural a secret. And seeing that some of the summoned Servants in this conflict are EGREGIOUSLY powerful (at least for the power levels in play here), this is easier said than done.
Though it is pretty difficult to blow this particular masquerade open (in canon, a Servant summoned Cthulhu in front of the city and they managed to hush it up), Bad Shit of a 100% lethal nature is guaranteed to happen if somebody goes even an hair too crazy, so teams are advised to play it as discretely as they can.
Whoever gets to make the wish, of course.
And, if no team manages it, whoever is believed to make it closer to the goal line.
If you believe that they both are equally likely to make it exactly the same distance, just give your nod to whoever you think will leave the smaller body count in getting there.