  We present a practical analysis of the fermion sign problem in fermionic path
integral Monte Carlo (PIMC) simulations in the grand-canonical ensemble (GCE).
As a representative model system, we consider electrons in a $2D$ harmonic
trap. We find that the sign problem in the GCE is even more severe than in the
canonical ensemble at the same conditions, which, in general, makes the latter
the preferred option. Despite these difficulties, we show that fermionic PIMC
simulations in the GCE are still feasible in many cases, which potentially
gives access to important quantities like the compressiblity or the Matsubara
Greens function. This has important implications for contemporary fields of
research such as warm dense matter, ultracold atoms, and electrons in quantum
dots.
