  Using a sample of $(448.1~\pm~2.9)$ $\times 10^{6}$ $\psi(3686)$ decays
collected with the BESIII detector at BEPCII, we report an observation of
$\Xi^{-}$ transverse polarization with a significance of $7.3 \sigma$ in the
decay $\psi(3686)\rightarrow\Xi^{-}\bar\Xi^{+}$
($\Xi^{-}\rightarrow\pi^{-}\Lambda$,
$\bar\Xi^{+}\rightarrow\pi^{+}\bar\Lambda$, $\Lambda\to p\pi^{-}$,
$\bar\Lambda\to\bar{p}\pi^{+}$). The relative phase of the electric and
magnetic form factors is determined to be $\Delta\Phi = (0.667 \pm 0.111 \pm
0.058)$ rad. This is the first measurement of the relative phase for a
$\psi(3686)$ decay into a pair of $\Xi^{-}\bar\Xi^{+}$ hyperons. The $\Xi^{-}$
decay parameters ($\alpha_{\Xi^{-}}$, $\phi_{\Xi^-}$) and their conjugates
($\alpha_{\bar\Xi^{+}}$, $\phi_{\bar{\Xi}^{+}}$), the angular-distribution
parameter $\alpha_{\psi}$, and the strong-phase difference
$\delta_{p}-\delta_{s}$ for $\Lambda$-$\pi^-$ scattering are measured to be
consistent with previous BESIII results.
