  In this paper, we study quantization on a compact integral symplectic
manifold $X$ with transversal real polarizations. In the case of complex
polarizations, namely $X$ is K\"ahler equipped with transversal complex
polarizations $T^{1, 0}X, T^{0, 1}X$, geometric quantization gives $H^0(X,
L^{\otimes k})$'s. They are acted upon by $\mathcal{C}^\infty(X, \mathbb{C})$
via Toeplitz operators as $\hbar = \tfrac{1}{k} \to 0^+$, determining a
deformation quantization $(\mathcal{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$
of $X$.\par We investigate the real analogue to these, comparing deformation
quantization, geometric quantization and Berezin-Toeplitz quantization. The
techniques used are different from the complex case as distributional sections
supported on Bohr-Sommerfeld fibres are involved.\par By switching the roles of
the two real polarizations, we obtain Fourier-type transforms for both
deformation quantization and geometric quantization, and they are compatible
asymptotically as $\hbar \to 0^+$. We also show that the asymptotic expansion
of traces of Toeplitz operators realizes a trace map on deformation
quantization.
