Instructions
Find a Phoenix Pinion in Alexandria the first time you're there. Check the wagon in the upper right part of the town square for one. The Phoenix Pinion will allow Eiko to summon the eidolon Phoenix when she joins your party later on.
Solve Ramuh's puzzle during the events at Pinnacle Rock to obtain a Peridot. The Peridot will allow Princess Garnet to summon the eidolon Ramuh.
Check Eiko's equipment when she joins your party during the events at Mountain Pass. She already has a Sapphire equipped; this allows her to summon the eidolon Fenrir.
Travel to the Iifa Tree. When Eiko breaks the seal, you will receive a Ruby. With the Ruby equipped, Eiko will be able to summon the eidolon Carbuncle.
Search for Princess Garnet while Brahne and Kuja fight during the events of your second visit to the Iifa Tree. Garnet will seek the aid of the imprisoned eidolon and receive an Aquamarine; this allows her to summon Leviathan.
Receive the Opal, Topaz and Amethyst from Doctor Tot during the events of Alexandria after the start of Disc Three. The Opal, Topaz and Amethyst teach Princess Garnet to summon the eidolons Shiva, Ifrit and Atomos, respectively.
Buy a Dark Matter from the Treno Auction House after you head there with Doctor Tot at the start of Disc Three. The Dark Matter will teach Garnet to summon the eidolon Odin.
Obtain the Ribbon during the events at Mt. Gulug. With the Ribbon, Eiko can summon the mighty eidolon Madeen.
Receive the Garnet (the stone, not the princess) from Beatrix once the party returns to Alexandria after the events in Mt. Gulug. With the Garnet (the stone), Garnet (the princess) can learn to summon the eidolon Bahamut.
Use Hades's synthesis shop in Memoria near the end of the game to forge a Pumice out of two Pumice Piece items. Using the Pumice, Garnet can learn to summon Ark, the grand eidolon. The first Pumice Piece you'll find is a reward for defeating Ark when you fight him in Oeilvert. The second Pumice Piece can be found as the reward for the "Outer Island 2" chocograph.