Let W be a scalar Hadamard-bisolution for the wave equation on a globally hyperbolic Lorentzian manifold M, then positivity W[φ,φ]≥0 only affects the symmetric part Ws of W, which is essentially given by the difference of Feynman and Anti-Feynman-propagator. Therefore Theorem 6.6.2 of  can be applied, which ensures the existence of some f∈C∞(M×M), s.t. Ws+f is a positive, symmetric bidistribution, but in general not a bisolution anymore. We will establish a well-posed Cauchy problem for symmetric bisolutions and use Ws+f as initial value in order to construct a global bisolution, from which we then show positivity.
 Duistermaat, J. J., Hörmander, L. - Fourier Integral Operators II, Acta Mathematica 128, 183-269 (1972)