  Most implementations of Coulomb-distorted strong-field approaches that
contain features such as tunneling and quantum interference use real
trajectories in continuum propagation, while a fully consistent approach must
use complex trajectories throughout. A key difficulty in the latter case are
branch cuts that appear due to the specific form of the Coulomb potential. We
present a method for treating branch cuts in quantum-trajectory models, which
is subsequently applied to photoelectron holography. Our method is not
numerically intensive, as it does not require finding the location of all
branching points and branch cuts prior to its implementation, and is applicable
to Coulomb-free and Coulomb-distorted trajectories. We show that the presence
of branch cuts leads to discontinuities and caustics in the holographic fringes
in above-threshold ionization (ATI) photoelectron angular distributions (PAD).
These artefacts are removed applying our method, provided they appear far
enough from the polarization axis. A comparison with the full solution of the
time-dependent Schr\"odinger equation is also performed, and a discussion of
the applicability range of the present approach is provided.
