If you don't know anyone who has flagged this quest, this is probably the most reliable - if lengthy - way to complete it, as she can give you a different set of cards when you go back to see her. Thus, it's entirely possible to collect the four different cards this way. Discarding the extra cards from a set may increase the odds of her giving you a different set the next time you speak with her.
Find other players who have started the quest and trade with them until you have all four different cards (Death, Hermit, King, Fool).
Once this quest is completed, it is possible to ask Chululu for your compatibility with another player. The player name is asked to be specified in a "Password" field on the screen, and the player must be nearby.
(Enter the name exactly as is in-game, the first letter capitalized. If you don't do this it says "too far away.")
Note: If you type Chululu's name for the compatibility, she'll reply "What? With me!? Spare me." If you type Kurou-Morou, she'll reply "What? With that creep? Over my dead body!" If you type your character's name she'll reply "Of course you're compatible with yourself! Hello!?"
You must gather one of each kind of Tarut card before your fortune can be told.
SPOILER WARNING: Click here for additional details about this story line.
A Real Collection
During Fan Festival 2008, attendees were given a package of Tarut Cards as part of their Gobbie Bags. Each package contained 5 copies of either The Fool, The Hermit, The King or Death. Just like the quest in game, you were to trade with your fellow players to obtain a full set of these cards (displayed below). These cards featured original artwork by Fumio Minagawa.