  We present a systematic exploration of the loss of predictivity in
Einstein-scalar-Gauss-Bonnet (ESGB) gravity. We first formulate a gauge
covariant method of characterizing the breakdown of the hyperbolicity of the
equations of motion in the theory. With this formalism, we show that strong
geodesic focusing leads to the breakdown of hyperbolicity, and the latter is
unrelated to the violation of the null convergence condition. We then
numerically study the hyperbolicity of the equations during gravitational
collapse for two specific ESGB gravity theories: "shift symmetric Gauss-Bonnet
gravity" and a version of the theory that admits "spontaneously scalarized"
black holes. We devise a "phase space" model to describe the end states for a
given class of initial data. Using our phase space picture, we demonstrate that
the two theories we consider remain predictive (hyperbolic) for a range of GB
couplings. The range of couplings, however, is small, and thus, the presence of
"spontaneously scalarized" solutions requires fine-tuning of initial data. Our
results, therefore, cast doubt as to whether scalarized black hole solutions
can be realistically realized in Nature even if ESGB gravity happened to be the
correct gravitational description.
