  For the first time, the strong phase difference between $D^0$ and
$\bar{D}^0\to\pi^+\pi^-\pi^+\pi^-$ amplitudes is determined in bins of the
decay phase space. The measurement uses $818\,\mathrm{pb}^{-1}$ of $e^+e^-$
collision data that is taken at the $\psi(3770)$ resonance and collected by the
CLEO-c experiment. The measurement is important for the determination of the $C
P$-violating phase $\gamma$ in $B^{\pm}\to D K^{\pm}$ (and similar) decays ,
where the $D$ meson (which represents a superposition of $D^0$ and $\bar{D}^0$)
subsequently decays to $\pi^+\pi^-\pi^+\pi^-$. To obtain optimal sensitivity to
$\gamma$, the phase space of the $D \to \pi^+\pi^-\pi^+\pi^-$ decay is divided
into bins based on a recent amplitude model of the decay. Although an amplitude
model is used to define the bins, the measurements obtained are
model-independent. The $CP$-even fraction of the $D \to \pi^+\pi^-\pi^+\pi^-$
decay is determined to be $F_{+}^{4\pi} = 0.769 \pm 0.021 \pm 0.010$, where the
uncertainties are statistical and systematic, respectively. Using simulated
$B^{\pm}\to D K^{\pm}, D \to \pi^+\pi^-\pi^+\pi^-$ decays, it is estimated that
by the end of the current LHC run, the LHCb experiment could determine $\gamma$
from this decay mode with an uncertainty of $(\pm10\pm7)^\circ$, where the
first uncertainty is statistical based on estimated LHCb event yields, and the
second is due to the uncertainties on the parameters determined in this paper.
