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The reason why a Catholic bishop's Coat of Arms are green dwells in the fact that green was the original color worn by bishops and archbishops was green, not purple.
The shield is the green ecclesiastical hat called a “gallero” with twenty tassels pendant on both sides. This broad brimmed hat, once worn in cavalcades, is no longer used but remains as a heraldic emblem. The original color worn by bishops and archbishops was green, not purple. This “episcopal color” is retained in heraldry. These external ornaments are those used for a prelate with the rank of Archbishop according to the Instruction of the Holy See, Ut Sive, of March, 1969. - Armorial Bearings of the Most Reverend Thomas Rodi, DD Ninth Bishop and Second Metropolitan Archbishop of Mobile, Alabama
The original color for the cassocks of bishop and archbishops was green. In the 16th century it was changed to purple.
For most of Church history green was the color for bishops. This color is still seen on the traditional coat of arms that each bishop chooses when elected. However, in the 16th century the color was switched to “amaranth red,” which is named so in reference to the color of the amaranth flower. In practicality it more resembles the color fuschia.
Since it is a color similar to purple, it holds a symbolic value that points to the task of the bishop to govern his local diocese. - How to tell the difference between a monsignor, a bishop and a cardinal
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Purple is the color of the Bishop's clothes only, and of the livery-garments at his ecclesiastical household. The true “episcopal color"—the one to be used for decoration purposes— is not purple, but green.
Taken from the footnote in Costume of Prelates of the Catholic Church by J.A.Nainfa