  We theoretically study an optical cavity and a $\mathcal{PT}$-symmetric pair
of mechanical resonators, where all oscillators are pairwise coupled, forming
an optomechanical system with a closed-contour interaction. Due to the presence
of both gain and feedback, we first explore its stability and the root loci
over a large coupling range. Under the red-sideband pumping, and for the
so-called $\mathcal{PT}$-unbroken phase it displays a double optomechanically
induced transparency for an experimentally realizable parameter set. We show
that both the transmission amplitude and the group delay can be continuously
steered from lower transmission window to the upper one by the phase of the
loop coupling parameter which acts as a static synthetic gauge field. In the
$\mathcal{PT}$-unbroken phase both the gain-bandwidth and delay-bandwidth
products remain constant over the full range of the controlling phase.
Tunability in transmission and bandwidth still prevails in the
$\mathcal{PT}$-broken phase. In essence, we suggest a simple scheme that grants
coupling phase dependent control of the single and double OMIT phenomena within
an effective $\mathcal{PT}$-symmetric optomechanical system.
