Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(I,X)= \{(x_i)\in \ell_\infty(I,X) : \lim_\mathcal{U}\|x_i\|=0\}$. We denote by $[x_i]$ the element of $X_\mathcal{U}$ which has $(x_i)\in \ell_\infty(I,X)$ as a representative.
There exists a natural embedding $X\to X_\mathcal{U}$ given by $x\to [x]$, where $(x)$ is the constant family, and using the principle of local reflexivity, we can construct a set $I_0$ and an ultrafilter $\mathcal{U}_0$ on $I_0$ so that the embedding $X\to X_{\mathcal{U}_0}$ extends to an embedding $X^{**}\to X_{\mathcal{U}_0}$. See the classical paper by S. Heinrich. Ultraproducts in Banach space theory. J. reine angew. Math. 313 (1980) 72-104.
When $E$ is a Banach lattice, the ultrapower $E_\mathcal{U}$ is also a Banach lattice, and the order in $E_\mathcal{U}$ induces through $E\to E_\mathcal{U}$ the order in $E$.
Question: Is it possible to find $I_0$ and $\mathcal{U}_0$ so that the order in $E_{\mathcal{U}_0}$ induces through $E^{**}\to E_{\mathcal{U}_0}$ the order in $E^{**}$?
I think that we would need a variation of the principle of local reflexivity respecting the order structure, but I do not know if this variation can be found in the literature.
