At higher loops it certainly isn't true that amplitudes are real. By the optical theorem the imaginary part of an amplitude is related to the intermediate states that can go on-shell. At higher loops lots of intermediate particles can go on shell so one gets an interesting imaginary part.

But here's one interesting way of getting a phase just at tree level. At tree level one would conclude from the optical theorem that there's no phase except at discrete momenta where the Mandelstam variables are equal to some mass-squared in the theory. But suppose we have a theory with an infinite tower of particles with increasing masses, with some small spacing between the masses. Then if we average over momentum scales that are large compared to this spacing, we could get a large imaginary part.

The classic example is high energy, small angle scattering in flat space string theory (or QCD), where the four-point amplitude (averaged over a range of energies that's much larger than the string scale) is

\(\Gamma(-1-\alpha' t)e^{-i\pi t}s^{\alpha' t},\)

where \(\alpha' \) is the Regge slope. The imaginary part is related to the fact that there is an infinite number of resonances that can be produced in the s-channel.

Another way you can get a phase in field theory is if there is a time delay. For example consider quantum field theory in the curved background of a highly boosted particle (a shockwave). In this background there is the Shapiro time-delay, see http://arxiv.org/pdf/1407.5597.pdf for a recent discussion. At large impact parameter and high energy (the eikonal approximation) one can resum the loops to get the answer

\(\exp(iG_{\text{N}}s\log b)\)

and differentiating with respect to the energy gives the standard answer for the Shapiro time-delay.