Defining particles by their commutation/anti-commutation relations

In my studies of many-body physics, I have encountered three types of particles, which can be defined based on their commutation/anti-commutation relations.

1. Fermions, defined by raising/lowering operators $a_i, a_j^dagger$ which anticommute on different sites:
   $$ {a_i^dagger, a_j} = delta_{ij} quad {a_i, a_j} = {a_i^dagger, a_j^dagger} = 0,$$

2. Bosons, defined by raising/lowering operators $b_i, b_j^dagger$ which commute on different sites:
   $$ [b_i, b_j^dagger] = delta_{ij} quad [b_i, b_j] = [b_i^dagger, b_j^dagger] = 0,$$

3. Spins (particularly, $1/2$), defined by raising/lowering operators $sigma^i_pm = sigma^i_x pm isigma^i_y,$ which obey fermionic relations on the same site and bosonic relations on different sites:
   $$ {sigma_i^+, sigma_i^-} = 1 quad [sigma_i^+, sigma_j^-] = 0, inot=j.$$

Clearly, based on this, one can consider the existence of a fourth type of particle, which is defined by raising/lowering operators $x_i, x_j^dagger$ which obey bosonic relations on the same site, and fermionic relations on different sites:
$$ [x_i, x_i^dagger] = 1 quad {x_i, x_j^dagger} = 0, inot=j.$$

Does such a particle exist? If not, is there a physical/mathematical reason that forbids such a particle from existing?