  We study theoretically an optical cavity and a parity-time
($\mathcal{PT}$)-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 explore its
stability and the root loci over a wide coupling range. Under the red-sideband
pumping and for the so-called $\mathcal{PT}$-unbroken phase it displays a
double optomechanically induced transparency (OMIT) for an experimentally
realizable parameter set. We show that both the transmission amplitude and the
group delay can be continuously steered from the lower transmission window to
the upper one by the loop coupling phase which breaks the time-reversal
symmetry and introduces 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, albeit over a reduced range. 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.
