Phase-isometries between normed spaces

Let $X$ and $Y$ be real normed spaces and $f \colon X\to Y$ a surjective mapping. Then $f$ satisfies $\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\} = \{\|x+y\|, \|x-y\|\}$, $x,y\in X$, if and only if $f$ is phase equivalent to a surjective linear isometry, that is, $f=σU$, where $U \colon X\to Y$ is a surjective linear isometry and $σ\colon X\to \{-1,1\}$. This is a Wigner's type result for real normed spaces.