So... is Full Moon wo Sagashite finished or not?
It is a bit ambiguous in this case.
Why?
The anime finished before the manga:
Anime finished: March 29, 2003
Manga finished: April 30, 2004
MangaUpdates contains information about the anime adaptation and indicates that it is complete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Mitsuki, a young girl with throat cancer, dreams of becoming a singer to reach her faraway friend, Eichi. Her grandmother, opposed to music, wants her to undergo surgery, but Mitsuki fears losing her voice.
Suddenly, she encounters two shinigami, Meroko and Takuto, who reveal she only has a year to live. This shocking news motivates her to pursue her dream with newfound urgency.
With the shinigami's assistance, Mitsuki strives to become a professional singer before her time runs out, exploring the meaning of music and life along the way. Full Moon wo Sagashite is her journey to achieve that dream against the odds.