In reading a recent preprint  contrasting bosonic models with local (tensor product) Hilbert spaces with SYK-like models of fermions, I realized I was confused about something. While I have a vague understanding that fermions are fundamentally non-local degrees of freedom (because of the non-trivial exchange statistics), I was surprised to read that the Hilbert space of the SYK model is non-local — that is, not a tensor product of local Hilbert spaces.
Naively, I would expect one could write down the Hilbert space as a tensor product of Majorana Hilbert spaces, just as in a spin model one uses a tensor product of spins. I guess this is complicated for Majorana fermions because each Majorana is somehow "half of a degree of freedom" and does not exist independently. Actually, I am not even sure how to write down the Hilbert space of a single Majorana fermion (or if that is even a notion that makes sense). I wonder if this complication could be removed by using Dirac fermions instead of Majoranas.
My specific questions are:
Q1) Can someone provide an explicit description of the non-local Hilbert space of a collection of Majorana fermions, as in the SYK model?
Q2) Is the non-locality (lack of tensor product structure) a consequence of using Majorana fermions, or would it also appear in the case of Dirac fermions?
 C. L. Baldwin, B. Swingle, Quenched vs Annealed: Glassiness from SK to SYK, https://arxiv.org/abs/1911.11865