Assume That $U,V$ are two filters on the natural number $\mathbb{N}$. We say that $U$ is equivalent to $V$ if there is a bijection $\phi: \mathbb{N} \to \mathbb{N}$ such that $\tilde{\phi}(U)=V$ where $\tilde{\phi}:P(\mathbb{N}) \to P(\mathbb{N})$ is the natural extension of $\phi$ to the power set $P(\mathbb{N})$. Let $U,V$ be two non principal ultra filter on $\mathbb{N}$.

Let $\mathbb{R}^*_{U}$ and $\mathbb{R}^*_{V}$ be the corresponding nonstandard extension of real numbers associated with $U$ and $V$, respectively.

Assume that $\mathbb{R}^*_{U}$ and $\mathbb{R}^*_{V}$ are isomorphic as fields. Does this imply that $U$ and $V$ are equivalent filters?

My apology in advance, if the question is elementary. The question arose me about 17 years ago when I was trying to understand the application of non standard analysis to ordinary differential equations.