When one studies quantum mechanics of more than one particle, one learns that all fundamental particles can be classified as either bosonic or fermionic. Fermions have a spinor structure, and pick up a phase of -1 when rotated by 2$\pi$ radians, while the (gauge) bosons have a vector nature and pick up a phase of 1. Mathematically, one says that spinors are a representation of the group SU(2), which is a double cover of the rotation group SO(3).
The standard way it is shown to undergraduates that something can rotate by 2$\pi$ and not end up in its initial configuration is the Dirac Belt trick, which essentially involves rotating one part of a belt while another point is fixed. Or, similarly, one can rotate a plate in one's hand while keeping one's body fixed (the Feynman Plate Trick).
One striking aspect of this demonstration, unlike engineering a physical representation of a system with 2$\pi$, $\pi$, or lower rotation symmetry, is that it depends essentially on the object's relation to its environment, in the sense that the connections to the environment (such as the body of the belt) are what prevent the system from having 2$\pi$ symmetry.
One known way of relating a system of (free) fermions to a system of (interacting) bosons is the Jordan-Wigner transformation. This transformation is, famously, highly nonlocal, in that a fermionic operator at one point is created by a long chain of the bosonic operators. This and related ideas have inspired Prof. Xiao-Gang Wen, in his textbook on many-body theory, to claim that fermions should be considered non-local excitations. However, I'm not aware of anyone else emphasizing this viewpoint.
Okay, with all that background established I have two questions:
Are fermions non-local objects, in a sense in which gauge bosons are not?
Is the fact that our physical representations of SU(2) symmetry, such as the Belt Trick, require some sort of connection to a fixed background a reflection of this fundamental non-locality, or a mere coincidence? Can this relationship be made more precise?
Edit: Thanks to all who contributed. It does not seem there is much consensus on what relation, if any, there is between SU(2) symmetry and non-local properties. I will give the bounty to the top-voted answer but consider the question not fully resolved and welcome any additional answers.