When it comes to finding synonyms for the phrase "deep dyed," it is important to consider the context in which it is being used. Generally speaking, the term refers to something that is inherently ingrained, whether that be a belief, emotion, or characteristic. Some synonyms for "deep dyed" that capture this idea include "inherent," "innate," and "ingrained." Other potential synonyms might focus on the intensity of the trait being described, such as "unwavering," "steadfast," or "unshakable." Whatever term one chooses to use, it is important to ensure that it captures the essence of the original phrase and conveys the intended meaning effectively.