  We say that a mapping $f: X \rightarrow Y$ between two real normed spaces is
a phase-isometry if it satisfies the functional equation \begin{eqnarray*}
\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \quad (x,y\in
X).\end{eqnarray*} A generalized Mazur-Ulam question is whether every
surjective phase-isometry is a multiplication of a linear isometry and a map
with range $\{-1, 1\}$. This assertion is also an extension of a fundamental
statement in the mathematical description of quantum mechanics, Wigner's
theorem to real normed spaces. In this paper, we show that for every space $Y$
the problem is solved in positive way if $X$ is a smooth normed space, an
$\mathcal{L}^{\infty}(\Gamma)$-type space or an $\ell^1(\Gamma)$-space with
$\Gamma$ being an index set.
