Phase-isometries between normed spaces

Let $X$ and $Y$ be real normed spaces and $f\text{\hspace{0.17em}⁣}:X\to Y$ a surjective mapping. Then $f$ satisfies ${|f(x)+f(y)|, |f(x)-f(y)|} = {|x+y|, |x-y|}$,$x,y\in X$,ifandonlyif$f$ is phase equivalent to a surjective linear isometry, that is, $f=\sigma U$, where $U\text{\hspace{0.17em}⁣}:X\to Y$ is a surjective linear isometry and $\sigma \text{\hspace{0.17em}⁣}:X\to \{-1,1\}$. This is a Wigner’s type result for real normed spaces.