  We prove that the entropy function on the moduli space of real quadratic
rational maps is not monotonic by exhibiting a continuum of disconnected level
sets. This entropy behavior is in stark contrast with the case of polynomial
maps, and establishes a conjecture on the failure of monotonicity for bimodal
real quadratic rational maps of shape $(+-+)$ which was posed in
arXiv:1901.03458 based on experimental evidence.
