The phrase "stubbornly as a mule" is used to describe someone who is persistent and unyielding in their beliefs, actions, or decisions. However, there are several synonyms that can be used to convey the same meaning. One such synonym is "inflexibly" which means unbending or unyielding. Another synonym is "obstinately" which means refusing to change one's behavior or ideas. "Tenaciously" is another synonym that means to hold fast or grip tightly. Lastly, "doggedly" means to pursue something with determination or persistence, which is similar to the unyielding nature of a mule.