So... is Ikoku Irokoi Romantan finished or not?
YES, it is finished.
Why?
The anime finished after the manga:
Anime finished: October 24, 2008
Manga finished: April 01, 2003
MangaUpdates contains information about the anime adaptation and indicates that it is complete.
Watching Order (by release date)
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Synopsis
Ranmaru, the heir to a Japanese yakuza clan, enters into a marriage of convenience aboard a cruise ship. However, a drunken night leads to an unexpected encounter with a handsome ship captain.
Their paths cross again in Italy, hinting at the blossoming of a passionate and dramatic romance amidst the backdrop of Ranmaru's complex life and the alluring allure of a forbidden love.