So... is Namiuchigiwa no Muromi-san finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: June 30, 2013
Manga finished: May 28, 2014
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
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
Namiuchigiwa no Muromi-san follows the mundane fishing life of Takurou, which is completely upended when he accidentally catches a mermaid named Muromi.
Muromi is a boisterous and energetic character who quickly latches onto Takurou, turning his peaceful fishing trips into chaotic adventures.
As Muromi introduces her quirky friends from the deep sea, Takurou's once quiet dock becomes a gathering spot for mythological creatures, much to his dismay.
The anime focuses on the humorous clash between Takurou's desire for a simple life and the increasingly bizarre situations he finds himself in thanks to Muromi and her friends.
Ultimately, it's a comedy about a regular guy trying to maintain some semblance of normalcy amidst a world of fantastical beings.