  Path integrals with complex actions are encountered for many physical systems
ranging from spin- or mass-imbalanced atomic gases and graphene to quantum
chromo-dynamics at finite density to the non-equilibrium evolution of quantum
systems. Many computational approaches have been developed for tackling the
sign problem emerging for complex actions. Among these, complex Langevin
dynamics has the appeal of general applicability. One of its key challenges is
the potential convergence of the dynamics to unphysical fixed points. The
statistical sampling process at such a fixed point is not based on the physical
action and hence leads to wrong predictions. Moreover, its unphysical nature is
hard to detect due to the implicit nature of the process. In the present work
we set up a general approach based on a Markov chain Monte Carlo scheme in an
extended state space. In this approach we derive an explicit real sampling
process for generalized complex Langevin dynamics. Subject to a set of
constraints, this sampling process is the physical one. These constraints
originate from the detailed-balance equations satisfied by the Monte Carlo
scheme. This allows us to re-derive complex Langevin dynamics from a new
perspective and establishes a framework for the explicit construction of new
sampling schemes for complex actions.
