  The characteristic initial (boundary) value problem has numerous applications
in general relativity (GR) involving numerical studies, and is often formulated
using Bondi-like coordinates. Recently it was shown that several prototype
formulations of this type are only weakly hyperbolic. Presently we examine the
root cause of this result. In a linear analysis we identify the gauge,
constraint and physical blocks in the principal part of the Einstein field
equations in such a gauge, and show that the subsystem related to the gauge
variables is only weakly hyperbolic. Weak hyperbolicity of the full system
follows as a consequence in many cases. We demonstrate this explicitly in
specific examples, and thus argue that Bondi-like gauges result in weakly
hyperbolic free evolution systems under quite general conditions. Consequently
the characteristic initial (boundary) value problem of GR in these gauges is
rendered ill-posed in the simplest norms one would like to employ. The
possibility of finding good alternative norms, in which well-posedness is
achieved, is discussed. So motivated, we present numerical convergence tests
with an implementation of full GR which demonstrate the effect of weak
hyperbolicity in practice.
