  Path-integral analyses originally pioneered in the study of the complex-phase
problem afflicting lattice calculations of finite-density quantum
chromodynamics are generalized to non-relativistic Fermi gases with repulsive
interactions. Using arguments similar to those previously applied to
relativistic theories, we show that the analogous problem in nonrelativistic
systems manifests itself naturally in Tan's contact as a nontrivial
cancellation between terms with varied dependence on extensive thermodynamic
quantities. We analyze that case under the assumption of gaussian phase
distribution, which is supported by our Monte Carlo calculations and
perturbative considerations. We further generalize these results to observables
other than the contact, as well as to polarized systems and systems with fixed
particle number. Our results are quite general in that they apply to repulsive
multi-component fermions, are independent of dimensionality or trapping
potential, and hold in the ground state as well as at finite temperature.
