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Is there any particular order I am supposed to find the phantom characters?
And, who am I supposed to be finding in the root towns?

I believe the order is.. Mimiru, Sora, BT, A-20? Not sure past BT.

the order is Mimiru, BT, Bear, Sora, Crim, SK, A-20, Subaru and Tsukasa

here is the list to where to find them and what you get: info was found at GameFAQ.com.

 .hack//Sign Ghost hunt... COLLECT THEM ALL!
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Ok, this little side-quest will be available ONLY after you see all the scenes
at Delta Hidden forbidden Holy Ground. IF you haven't I reccomend you do so.
The Sign characters will appear, only one at a time. IN THE FOLLOWING ORDER.
Happy hunting, y'all!
**KEY**
1=Delta Server
2=Theta Server
3=Lambda Server
4=Sigma Server
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Mimiru: ----GIFT------
1. Where you first met Mia in ally near Scroll shop. Healing Elixir
2. By Grunty Farm. Emperor's soul
1. Weapon shop. Noble Wine
3. Grunty Farm. Energy Sutras
4. On path from Elf's Haven to Grunty Farm. Spirit Sutras
2. Behind Save shop. Golden Grunty
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BT: ----GIFT------
3. Between the three shops at end of path. Healing Elixir
4. Between Weapon and Scoll shops. Emperor's Soul
2. By Scroll shop. Noble Wine
1. Item shop Energy Sutras
4. Between Item shop and Elf's Haven. Spirit Sutras
3. Grunty Farm. BT's Wand
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Bear: ----GIFT------
2. By Item, and Elf's Haven. Healing Elixir
1. In alley where Mimiru/Mia were. Emperor's Soul
3. Weapon shop. Noble Wine
4. Between Elf Haven and Save Shop. Energy Sutras
2. By Grunty Farm. Spirit Sutras
3. In Grunty Corral with Noble Grunty. Silver Grunty
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Sora: ----GIFT------
4. Between Grunty shop and Save shop. Healing Elixir
1. By Elf's Haven. Emperor's Soul
3. By Grunty Stable. Noble Wine
4. On path from chaos gate to magic shop. Energy Sutras
2. On path to grunty farm. Spirit Sutras
4. Behind Grunty Farm. Golden Grunty
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Crim: ----GIFT------
3. Walking between Save shop and Weapon shop Healing elixir
4.Sigma, between weapon and save shop: Emperor's Soul
2.Second server, by save shop pacing around: Noble Wine
1.First server, by magic shop: Energy Sutras
3. He walks a large area, he is easy to spot Spirit Sutras
4. Standing by magic shop in NE corner. Silver Grunty
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Silver Knight: ----GIFT------
1. Bridge to back of area. Healing Elixir
3. Outside grunty farm. Emperor's Soul
2. By Scroll and Weapon shop. Noble Wine
4. Walking path between Grunty Farm and Elf's Haven. Energy Sutras
2. By Potion shop. Spirit Sutras
1. Standing by Weapons shop. SK's Sword
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A-20: ----GIFT------
2. By grunty farm. Healing Elixir
1. In alley by Scroll shop. Emperor's Soul
4. On path between Weapon shop and Grunty farm. Noble Wine
3. Grunty Farm and Stable: Energy Sutras
1. Walks between Item shop and Elf's Haven. Spirit Sutras
4. Item shop, standing in NW corner. Golden Grunty
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Subaru: ----GIFT------
4.Behind Grunty Farm. Healing Elixir
1.Walks between Weapon and Scroll shops. Emperor's Soul
2. By Item shop and Elf's Haven. Noble Wine
4. Path between Item and Elf's Haven. Energy Sutras
3. Walks from Save shop to Weapons shop. Spirit Sutras
1. Scroll shop. Silver Grunty
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Tsukasa: ----GIFT------
1. Alleyway Healing Elixir
2. Back of Grunty Farm. Emperor's Soul
3. Between the Grunty Farm and Stable. Noble Wine
4. Path between Weapon shop and Grunty Farm. Energy Sutras
4. Between Elf's Haven and Grunty Farm. Spirit Sutras
2. Behind Grunty Farm. Tsukasa's Wand