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I found that placing my fairy in a tactical position will keep it alive in certain fights where the enemy targets you with devistating aoe. It is especially efective in the relic chimera and hydra fights. If you let it follow you, it will get hit by something and die.
But for say Ifrit/Titan fight it's slightly better for it to be on follow for you to avoid it taking the AoE damage then wasting a turn for it to heal itself.
For ifrit it would probably be best to have your fairy right next to ifrit the whole time. I don't think Ifrit will target it for eruptions. And your fairy will get hit by a plume/eruption if it is following you because it moves after you do, and the delay in movement gets it hit.
For Garuda, I have been placing my fairy right behind garuda. on the wind phase I just let it follow.