The metaplectic correction in geometric quantization

Let $P$ be a polarization on a symplectic manifold for which there exists a metalinear frame bundle. We show that for any other compatible polarization $P^{\mathrm{\prime }}$ there exists a unique metalinear frame bundle such that the BKS-pairing is well defined. This means that we do not need the metaplectic frame bundle (nor a positivity condition on $P$) to achieve this goal, and thus the name “metaplectic correction” is inappropriate.