  Given a real elliptic curve $E$ with non-empty real part and $[D]\in
\mbox{Pic}^2 E$ its $g_2^1$, we study the real inflection points of
distinguished subseries of the complete real linear series
$|\mathcal{L}_\mathbb{R}(kD)|$ for $k\geq 3$. We define {\it key polynomials}
whose roots index the ($x$-coordinates of) inflection points of the linear
series, away from the points where $E$ ramifies over $\mathbb{P}^1$. These fit
into a recursive hierarchy, in the same way that division polynomials index
torsion points.
  Our study is motivated by, and complements, an analysis of how inflectionary
loci vary in the degeneration of real {\it hyperelliptic} curves to a metrized
complex of curves with elliptic curve components that we carried out in our
previous article with Biswas.
