Phase-isometries between normed spaces

Abstract: 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, $f=\sigma U$, where $U \colon X\to Y$ is a surjective linear isometry and $\sigma \colon X\to {-1,1}$. This is a Wigner's type result for real normed spaces.