  We show that any SYK-like model with finite-body interactions among
\textit{local} degrees of freedom, e.g., bosons or spins, has a fundamental
difference from the standard fermionic model: the former fails to be described
by an annealed free energy at low temperature. In this respect, such models
more closely resemble spin glasses. We demonstrate this by two means: first, a
general theorem proving that the annealed free energy is divergent at low
temperature in any model with a tensor product Hilbert space; and second, a
replica treatment of two prominent examples which exhibit phase transitions
from an "annealed" phase to a "non-annealed" phase as a function of
temperature. We further show that this effect appears only at $O(N)$'th order
in a $1/N$ expansion, even though lower-order terms misleadingly seem to
converge. Our results prove that the non-bosonic nature of the particles in SYK
is an essential ingredient for its physics, highlight connections between local
models and spin glasses, and raise important questions as to the role of
fermions and/or glassiness in holography.
