  The detection of large angular scale $B$-mode in the Cosmic Microwave
Background (CMB) polarization signal will open a direct window into not only
the primary CMB anisotropies caused by the primordial gravitational waves (PGW)
originating in the epoch of inflation, but also the secondary anisotropies
imprinted during the epoch of cosmic reionization. The existence of patchiness
in the electron density during reionization produces a unique distortion in the
CMB $B$-mode polarization, which can be distinguished from the PGW signal with
the aid of spatial frequency modes. In this work, we employ an $EB$ estimator
by combining $E$-mode and $B$-mode polarization for the $\tau$ power spectrum
signal generated in a photon-conserving semi-numerical reionization model
called SCRIPT. We developed a Bayesian framework for the joint detection of the
PGW and reionization signal from CMB observations and show the efficacy of this
technique for upcoming CMB experiments. We find that, for our model, the $\tau$
power spectrum signal effectively tracks the inhomogeneous electron density
field, allowing for robust constraints on the patchy $B$-mode signal. Further,
our results indicate that employing the $EB$ estimator for the $\tau$ signal
will facilitate ground-based CMB-S4 to detect the patchy $B$-mode signal at
approximately $\geq 2\sigma$ confidence level while observations with
space-based PICO will improve this detection to $\geq 3\sigma$ going as high as
$\geq 7\sigma$ for extreme reionization models. These findings not only
highlight the future potential of these experiments to provide an improved
picture of the reionization process but also have important implications
towards an unbiased measurement of $r$.
