  We analyze real-time thermal correlation functions of conserved currents in
holographic field theories using the grSK geometry, which provides a contour
prescription for their evaluation. We demonstrate its efficacy, arguing that
there are situations involving components of conserved currents, or derivative
interactions, where such a prescription is, in fact, essential. To this end, we
first undertake a careful analysis of the linearized wave equations in AdS
black hole backgrounds and identify the ramification points of the solutions as
a function of (complexified) frequency and momentum. All the equations we study
are Fuchsian with only regular singular points that for the most part are
associated with the geometric features of the background. Special features,
e.g., the appearance of apparent singular points at the horizon, whence
outgoing solutions end up being analytic, arise at higher codimension loci in
parameter space. Using the grSK geometry, we demonstrate that these apparent
singularities do not correspond to any interesting physical features in
higher-point functions. We also argue that the Schwinger-Keldysh collapse and
KMS conditions, implemented by the grSK geometry, continue to hold even in the
presence of such singularities. For charged black holes above a critical
charge, the energy density operator does not possess an exponentially growing
mode, associated with `pole-skipping' (from one such apparent singularity). Our
analysis suggests that the connection between the scrambling physics of black
holes and energy transport has, at best, a limited domain of validity.
