Flan Princesses can only be found in one room of the Lunar Subterrane (the room has a single chest containing a Red Fang). In the Advance and Complete Collection ports a group of eight Flan Princesses (rather than the usual five in Lunar Subterrane) can be found in the Flan room of the Lunar Ruins. Sirens can be used to trigger the Subterrane formation, but sadly not the higher count group in the Ruins. They are harder to find, but have better odds of a tail drop by sheer numbers.
As soon as the fight begins, a Flan Princess will say "Dance with us!" and cast a party-wide Berserk spell called Samba de Flan. Occasionally, a Flan Princess will cast a spell called Entice to inflict the Confuse effect.
In the Flan room in the Lunar Ruins, after the first rare encounter of the eight Flan Princesses, they start to appear after every five or six random encounters. After five to seven of these encounters with the Flan Princesses, they will stop appearing. The player can quicksave, then reload the game to start it again. The player will have to spend time finding the rare encounter again, but they will start to appear commonly again after that. This makes acquiring the Pink Tail easier.
In most versions of the game, one should not have much trouble taking Flan Princesses down with high levels, good equipment, and magic/summons, but in the 3D remake version they are more difficult and will require a different strategy.
Before engaging them, the player should ensure everyone has their strongest weapons, with Cecil equipped with the Crystal Mail and a Headband (giving him the Counter and Draw Attacks Augments may also be a good idea). The player should ensure these weapons do not have an elemental affinity, excluding Holy and Dark.
While everyone else in the party will be affected by the Flan Princesses' Berserk, the Crystal Mail immunizes Cecil to the spell's effect. The party should cast Protect and resist the urge to have Cecil attack the Princesses as he will have his hands full keeping everyone alive. The Headband will protect Cecil from Confusion and with everyone healed, eventually the Princesses will die.
An alternate strategy for the 3D remake version of the game is to equip Cecil with the Draw Attacks and Counter Augments, place him in the back row, and place Stardust (a consumable item which casts Comet - it can be purchased at the item shop in the dwarf village of Tomra) in his top ability slot. Each time he is attacked, he will use Stardust, doing approximately 3,500 - 4,500 damage; at higher levels Cecil will have no trouble hitting over 7,000 damage. The battle should end in two or three hits. The only downside is the amount of Stardusts necessary to continually fight in this way - a full inventory of them will be used up in 33 or 49 battles, and they are expensive at 4,800 gil apiece; each siren used per battle costs an additional 3,000 gil.
Yet another strategy is to, using the Crystal Mail to give Cecil immunity to Berserk and equipping him with Draw Attacks, have him cast Esuna on Rydia. Rydia should have plenty of time to summonBahamut before either Cecil dies or the Princesses cast a second Samba de Flan, if Cecil has adequate Defense and HP. If Cecil also has Counter equipped, he will have already dealt some damage to each Flan Princess, and Bahamut is likely to deal an additional 9,999 damage to each, killing them. Placing Cecil in the back row with his most defensive equipment will help him last longer, as will having him use Defend or Brace once he has cast Esuna on Rydia.
An easier way without spending money on Stardust is to give Rydia the Counter Augment and, if the party has the Stardust Rod, have her put it in the first slot and Counter at the bottom. The player should cast the Esuna spell on her and the next time she's hit she'll counter using the rod and doing nigh 7,000 damage.