  We study the ferromagnetic Ising model on the infinite $d$-regular tree under
the free boundary condition. This model is known to be a factor of IID in the
uniqueness regime, when the inverse temperature $\beta\ge 0$ satisfies $\tanh
\beta \le (d-1)^{-1}$. However, in the reconstruction regime ($\tanh \beta >
(d-1)^{-\frac{1}{2}}$), it is not a factor of IID. We construct a factor of IID
for the Ising model beyond the uniqueness regime via a strong solution to an
infinite dimensional stochastic differential equation which partially answers a
question of Lyons. The solution $\{X_t(v) \}$ of the SDE is distributed as
  \[
  X_t(v) = t\tau_v + B_t(v),
  \] where $\{\tau_v \}$ is an Ising sample and $\{B_t(v) \}$ are independent
Brownian motions indexed by the vertices in the tree. Our construction holds
whenever $\tanh \beta \le c(d-1)^{-\frac{1}{2}}$, where $c>0$ is an absolute
constant.
