multiplying a function in $H^s$ times a function taking values in $S^1$

Question

If $\phi=\phi(x)$ is a function $\phi:\mathbb{R}^d\to\mathbb{C}$ in $H^s(\mathbb{R}^d)$ then it's clear from the definition of the inner product in $H^s(\mathbb{R}^d)$, \begin{equation} \langle f, g\rangle_{s}=\sum_{|\alpha| \leq s}\left\langle\partial_x^\alpha f, \partial_x^\alpha g\right\rangle_{L^2(\mathbb{R}^d)} \end{equation} that $e^{i\alpha}\phi$ is also in $H^s(\mathbb{R}^d)$ for any phase $\alpha\in\mathbb{R}$. However, is that the case if one allows $\alpha=\alpha(x)$ to be a (real-valued) function too? I've tried tackling this problem using the definition of $H^s(\mathbb{R}^d)$ in terms of Fourier transforms without any success: the Fourier transform of $e^{i\alpha(x)}\phi(x)$ doesn't have a nice expression in terms of the Fourier transform of $\phi(x)$, as far as I know. Ideally I would like to find a bound of the form $$ \|e^{i\alpha(x)}\phi(x)\|_{s}\lesssim \|\phi(x)\|_{s} $$ but I don't know if this is even true. If it helps, you can assume that $d=3$ and (if really necessary) that $s>d/2=3/2$. Any help would be much appreaciated, thank you!