  A simple integral relation between a complex weight and the corresponding
positive distribution is derived by introducing a second complex variable.
Together with the positivity and normalizability conditions, this sum rule
allows to construct explicitly equivalent pairs of distributions in simple
cases. In particular the well known solution for a complex gaussian
distribution is generalized to an arbitrary complex slope. This opens a
possibility of positive representation of Feynman path integrals directly in
the Minkowski time. Such construction is then explicitly carried through in the
second part of this presentation. The continuum limit of the new representation
exists only if some of the additional couplings tend to infinity and are tuned
in a specific way. The approach is then successfully applied to three quantum
mechanical examples including a particle in a constant magnetic field -- a
simplest prototype of a Wilson line. Further generalizations are shortly
discussed and an amusing interpretation of new variables is briefly mentioned.
