  The monotonicity of entropy is investigated for real quadratic rational maps
on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of
the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic,
covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like
mappings, we prove that the level sets of the real entropy function
$h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the
unimodal region in $\mathcal{M}_2(\mathbb{R})$. Based on the numerical
evidence, we conjecture that the monotonicity holds throughout the unimodal
region, but we conjecture that it fails in the region of $(+-+)$-bimodal maps.
