  Using a sample of $1.31\times10^9$ $J/\psi$ events collected with the BESIII
detector at the electron-positron collider BEPCII, we analyse the full $J/\psi
\to$ $\Xi^-\overline{\Xi}^+$, $\Xi^-\to \Lambda \pi^-$, $\Lambda\to p\pi^-$,
$\overline{\Xi}^+\to\overline{\Lambda}\pi^+$,
$\overline{\Lambda}\to\overline{p}\pi^+$ decay chain. A new method, exploiting
the fact that the $\Xi^-\overline{\Xi}^+$ pair is entangled and sequentially
decaying, and where the complete decay chains are reconstructed, is applied for
the first time. This enables precision measurements of the decay parameters for
the $\Xi^-\to\Lambda\pi^-$ decay ($\alpha_{\Xi}$, $\phi_{\Xi}$) as well as the
$\overline{\Xi}^+\to\overline{\Lambda}\pi^+$ decay ($\overline{\alpha}_{\Xi}$,
$\overline{\phi}_{\Xi}$). From the decay parameters, two independent CP tests
were performed, quantified by the observables $A_{\rm CP}^{\Xi}$ and $\Delta
\phi_\Xi$. Our results, $A_{\rm CP}^{\Xi}$ = $(6.0\pm13.4\pm5.6)\times10^{-3}$
and $\Delta \phi_\Xi= (-4.8\pm13.7\pm2.9)\times10^{-3}~{\rm rad}$, are
consistent with CP symmetry. Furthermore, our method enables a separation of
strong and weak $\Xi\to\Lambda\pi$ decay amplitudes. This results in the first
direct measurement of the weak phase difference for any baryon decay. The
result is found to be $(\xi_{P} - \xi_{S}) = (1.2\pm3.4\pm0.8)\times10^{-2}$
rad and is one of the most precise tests of CP symmetry for strange baryons.
The strong phase difference is measured to be $(\delta_P - \delta_S) =
(-4.0\pm3.3\pm1.7)\times10^{-2}$ rad. In addition, we provide an independent
measurement of the recently debated $\Lambda$ decay parameter,
$\alpha_{\Lambda} = 0.757 \pm 0.011 \pm 0.008 $. The
$\Lambda\overline{\Lambda}$ asymmetry is measured to be $A_{\rm CP}^{\Lambda} =
(-3.7\pm11.7\pm9.0)\times10^{-3}$.
