  We formulate a general multi-mode Gaussian operator basis for fermions, to
enable a positive phase-space representation of correlated Fermi states. The
Gaussian basis extends existing bosonic phase-space methods to Fermi systems
and thus enables first-principles dynamical or equilibrium calculations in
quantum many-body Fermi systems. We prove the completeness and positivity of
the basis, and derive differential forms for products with one- and two-body
operators. Because the basis satisfies fermionic superselection rules, the
resulting phase space involves only c-numbers, without requiring anti-commuting
Grassmann variables.
