  Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles,
i.e., unit squares with colored edges, that admit valid tilings (contiguous
edges of adjacent tiles have the same color) of the plane, none of them being
invariant under a nontrivial translation. We study herein the Wang shift
$\Omega_0$ made of all valid tilings using the set $\mathcal{T}_0$ of 11
aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a
minimal subshift $X_0$ of $\Omega_0$ such that every tiling in $X_0$ can be
decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112
that are equivalent to a set of 19 self-similar and aperiodic Wang tiles. We
suggest that this provides an almost complete description of the substitutive
structure of Jeandel-Rao tilings, as we believe that $\Omega_0\setminus X_0$ is
a null set for any shift-invariant probability measure on $\Omega_0$. The proof
is based on 12 elementary steps, 10 of which involve the same procedure
allowing one to desubstitute Wang tilings from the existence of a subset of
marker tiles. The 2 other steps involve the addition of decorations to deal
with fault lines and changing the base of the $\mathbb{Z}^2$-action through a
shear conjugacy. Algorithms are provided to find markers, recognizable
substitutions, and shear conjugacy from a set of Wang tiles.
