"Constrain then quantise" vs. "quantise then constrain" Consider a classical system whose configuration space is a manifold $M$, and which is subject to some constraint $\mathcal{C}=0$.
[E.g. the system could be a particle moving in $M=\mathbb{R}^n$, with $\mathcal{C}=x_1$ constraining its first coordinate to be zero. Or the system could be a gauge theory, with $M$ the space of connections on a time-slice, and $\mathcal{C}=\nabla\cdot\vec{E}$ the Gauss constraint.]
Let $\mathcal{P}\subset T^\ast M$ be the "reduced phase space" of the system, i.e. the submanifold of $T^\ast M$ on which the constraint is satisfied.
It seems there are (at least) two ways to quantise such a system:

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*Dirac Quantisation ("quantise then constrain"). First define an unphysical Hilbert space $\mathcal{H}= L^2(M)$. Then define the physical Hilbert space $\mathcal{H}_{dirac}\subset \mathcal{H}$ as the kernel of the constraint operator $\hat{\mathcal{C}}:\mathcal{H}\to\mathcal{H}$.


*Geometric Quantization ("constrain then quantise"). First choose a polarisation of $\mathcal{P}$, i.e. a foliation of $\mathcal{P}$ by Lagrangian submanifolds. Then choose any leaf $\mathcal{Q}$ of this foliation.  Finally, define the physical Hilbert space $\mathcal{H}_{geom}=L^2(\mathcal{Q})$.
I'm interested in the relation between the physical Hilbert spaces obtained using the two methods. If the state of the physical system is represented by some element of $\mathcal{H}_{geom}$ in geometric quantisation, what is the element of $\mathcal{H}_{dirac}$ that represents it in Dirac quantisation? Both Hilbert spaces are meant to describe the same physical system, after all.
I'd be satisfied with a broad-strokes description of how these two quantisation procedures are related. I'm fine with some mathematical details being swept under the rug, e.g. problems that occur in infinite dimensions.
 A: The phrase "quantization commutes with constraints" usually refers to Guillemin-Sternberg conjecture. It has only been proved for a limited class of (gauge) theories.
It should stressed that the word "constraints" here refer to a (gauge) symmetry, cf. the Dirac conjecture. It does not refer to polarization in geometric quantization.
See also this related Phys.SE post.
A: Take your first example of a particle in $\mathbb R^d$ constrained to lie on $x_1=0$. In Dirac Quantization the constraint on the wavefunctions is not $x_1 \psi(\vec x)=0$, but rather $$\frac{\partial}{\partial x_1}\psi(\vec x)=0.$$
This tells us that the 'physical states' in Dirac Quantization are constant in the $x_1$ direction. Therefore there is a simple map between physical states in Dirac Quantization and Geometric Quantization: $$\psi_{\text{dirac}}(x_1,...,x_d)=\psi_{\text{geometric}}(x_2,...,x_d).$$
To make the analogy with Gauge theories, the constraint $x_1=0$ is analogous to fixing the gauge: $\partial_i A^a_i=0$ (or whatever other gauge you like) and the constraint $\frac{\partial}{\partial x_1}=0$ is analogous to the Gauss constraint: $iD_i\frac{\partial}{\partial A^a_i}\equiv D_iE_i^a=0$ (recall $E_i^a\equiv i\frac{\partial}{\partial A^a_i}$ because it is canonically conjugate to $A^a_i$). (Also $D_i$ here is the covariant derivative.)
