I am asking this question as a mathematician trying to understand quantum theory, so please forgive my naivety.
Systems satisfying the canonical commutation relations are naturally modeled with symplectic geometry: for example, in the discrete setting, there is a deep connection between the stabilizer formalism and affine symplectic geometry.
Is there a analogous geometry which naturally models systems satisfying the anticommutation relations?
The algebra of a finite set of anticommuting $a_k$ and $a^\dagger_k$ is naturally connected with the orthogonal group. In particular the set $$ \gamma^{2n-1}=\hat a_n^{\dagger}+ \hat a_n, \nonumber\\ \gamma^{2n} = i(\hat a_n^\dagger-\hat a_n), $$ $n=1,\ldots N$ generates the Clifford algebra ${\rm Cl}_{2N}$.
More generally, in classical Hamiltonian formalism with supernumber-valued variables, the phase space is a supermanifold with a super-Poisson bracket $\{\cdot,\cdot\}_{SPB}$.
The super-Poisson bracket corresponds to a super-commutator $\frac{1}{i\hbar}[\cdot,\cdot]_{SC}$ upon quantization. The super-commutator is a commutator (an anticommutator) in the Grassmann-even (Grassmann-odd) sector, respectively.
In super-Darboux coordinates, the CCR (CAR) of the Grassmann-even (Grassmann-odd) coordinates is given by an antisymmetric (symmetric) matrix, leading to Weyl (Clifford) algebras, associated with with symplectic (indefinite orthogonal) symmetry groups, respectively.
