  We investigate the switching dynamics in a $\mathcal{PT}$-symmetric fiber
coupler composed of a saturable nonlinear material as the core. In such a
saturable nonlinear medium, bistable solitons may evolve due to the balance
between dispersion and saturable nonlinearity, which we extend in the context
of the $\mathcal{PT}$-symmetric coupler. Our investigations of power-controlled
and phase-sensitive switching show richer soliton switching dynamics than the
currently existing conventional counterparts, which may lead to ultrafast and
efficient all-optical switching dynamics at very low power owing to the
combined effects of $\mathcal{PT}$ symmetry and saturable nonlinearity. In
addition to the input power, the relative phase of the input solitons and
saturable coefficient are additional controlling parameters that efficiently
tailor the switching dynamics. Also, we provide a suitable range of system and
pulse parameters that would be helpful for the practical realization of the
coupler to use in all-optical switching devices and photonic circuits. Finally,
we develop a variational approach to analytically investigate the switching
dynamics in such $\mathcal{PT}$-symmetric couplers that excellently predicts
the numerical findings.
