  We introduce a positive phase-space representation for fermions, using the
most general possible multi-mode Gaussian operator basis. The representation
generalizes previous bosonic quantum phase-space methods to Fermi systems. We
derive equivalences between quantum and stochastic moments, as well as operator
correspondences that map quantum operator evolution onto stochastic processes
in phase space. The representation thus enables first-principles quantum
dynamical or equilibrium calculations in many-body Fermi systems. Potential
applications are to strongly interacting and correlated Fermi gases, including
coherent behaviour in open systems and nanostructures described by master
equations. Examples of an ideal gas and the Hubbard model are given, as well as
a generic open system, in order to illustrate these ideas.
