The Elemental Archfiends are bosses in the 3D versions of Final Fantasy IV fought in the Giant of Babil. Prior to the fight, the Archfiends were revived by Zemus so they could give one last group attack upon Cecil Harvey and his party.
The bosses are fought one after the other. The fiends are fought in the same order they were first encountered; Scarmiglione, Cagnazzo, Barbariccia, and Rubicante.
Cagnazzo is weak to Ice normally, but absorbs it while surrounded by water. During which, he becomes weak to Lightning instead. While surrounded by water, Cagnazzo can cast Tsunami on the whole party.
Barbariccia is immune to physical attacks due to the absence of Kain and his Jump ability, but magical attacks are capable of harming her. Barbariccia does not counter attacks as much, but she does attack more frequently than before. Though it is not mentioned in the bestiary, Holy magic is incredibly effective against her.
Rubicante can once again open his cloak and cast Inferno, which will inflict massive damage to the party while healing him for a fair bit of health. He also counters physical-based commands with Firaga on the whole party, and again casts Blizzara on himself when hit by one of Rydia's Summons.
Scarmiglione likely recasts any debuffs the player has restored. Using Ifrit or Firaga from Rydia and/or Firaga from Fusoya are good, but at the same time unwise due to the potential Gas counter. Fusoya can also heal along with Rosa.
Against Barbariccia Edge and Cecil have little to contribute offensively, so Rydia and Fusoya should focus on dealing out damage while Cecil and Rosa keep everyone's health up. Fusoya can cast Holy or Rydia can summon Dragon. If Rosa is a high enough level, she can Dualcast Holy dealing approximately 19998 damage.
When up against Rubicante, the player can use the same strategies as in the first encounter, except this time Fusoya replaces Kain—he works well as a second Black Mage (casting Blizzara and Blizzaga, for example) or as an additional White Mage if needed. When Rubicante opens his cloak, Edge should use Steal to force him to close it. If the player has completed the sidequest in the Feymarch and have access to Leviathan they should summon him here, as it is particularly devastating against Rubicante and can easily cause 9,999 HP of damage.
An alternate strategy is to cast Haste on Rydia, then use Shiva or Leviathan when Rubicante opens his cloak (more preferable with Shiva as she is faster), then use Steal with Edge. After around four turns of this, Rubicante should fall. Rosa can either revive characters killed by Rubicante's physical attack, or heal them if they get too injured by it.