# Connection between Poisson bracket and Anti-commutator?

Canonical quantization promotes Poisson brackets in classical mechanics to commutators in quantum mechanics. Is there any classical counterpart similar to the Poisson bracket for the anticommutator?

FWIW, in classical Hamiltonian formalism, 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.