Can two different complexity classes be equal relative to an oracle? Example request
Let $\mathsf{C}$ be a complexity class of your choice, let $\mathsf{O}$ be an oracle of your choice, and define $\mathsf{D} = \mathsf{C}^\mathsf{O}$. Then $\mathsf{C}^\mathsf{O} = \mathsf{D}^\mathsf{O}$, but it could happen that $\mathsf{C} \neq \mathsf{D}$. For example, you can take $\mathsf{C} = \mathsf{P}$ and $\mathsf{O} = \mathsf{HALT}$, the halting problem.

This might seem like cheating, but it's exactly the same answer as the one by Willard Zhan.