quantum field theory - Feynman paths of FTL velocity have imaginary momentum?

In this Phys.SE answer it is discussed that Feynman path integrals sums amplitudes for all possible paths, including those that are not time-like. If you take the momentum-space path integrals, I would naively expect that such space-like coordinate paths would contribute an imaginary momentum in the momentum-space path integral, that would result in evanescent exponentially decaying amplitudes outside the light-cone.

Is this a correct interpretation and/or expectation to have? furthermore, In the momentum propagator integrals discussed in the lecture notes Feynman Diagrams for Beginners (PDF) by Kresimir Kumericki, the integrands are $d^3 k$, so it seems that they cover the whole 3D real momentum space, which seems to me to be restricted inside the light-cone (by nature of it being real). This confuses me because it seems to contradict the idea that all paths are included in the integral. What am I missing?