dear Ettore, thanks for your question.
Let us restate it: at first order, the field Psi coincides with the perturbations phi, and in this sense, both of them must satisfy hyperbolic boundary conditions, if the Minkowski metric has emerged. This might seem contradictory with the fact that the field Psi a priori is living in a euclidian space and therefore should have some elliptic boundary conditions.
There is in fact no contradiction:
If one considers the full theory, that is we have nonlinear kinetic terms, there is a mismatch between signature of the metric and hyperbolicity-ellipticity, the latter being determined by a matrix involving the metric AND the derivatives of the fields. More precisely, the signature of the metric tensor determines the hyperbolicity-ellipticity only for canonical kinetic terms, for which the matrix in front of the second derivatives in the PDE is just the metric tensor. For nonlinear kinetic terms (our case), this is not true.
As a consequence, when dealing with the full theory, the chosen boundary conditions can be elliptic or hyperbolic, but when restricted to the (linear) operator constructed at *first* order (so that everything becomes linear), they can definitely be interpreted as hyperbolic boundary conditions, and there is no mismatch.
Hope this helps!
Florian, Stefano, Lorenzo
