This should be easy to prove if it is true, but, alas, what SHOULD be easy is not always easy for me ;)
Conjecture 1. Let $X$ be a real Banach space and let $X_\mathbb{C}$ denote its complexification. Suppose $(x_n\oplus iy_n)$ is a normalized basic sequence in $X_\mathbb{C}$. Then either $(x_n)$ or $(y_n)$ contains a basic subsequence in $X$.
It seems like it should be true, especially since a similar statement holds for seminormalized weakly null sequences. But maybe there is a counterexample.
Or, if the above is not true, the following weaker conjecture (together with Rosenthal's $\ell_1$ Theorem) would suffice for my purposes.
Conjecture 2. Let $X$ be a real Banach space and let $X_\mathbb{C}$ denote its complexification. Suppose $(x_n\oplus iy_n)$ is a normalized basic sequence in $X_\mathbb{C}$, which is equivalent to the canonical basis for $\ell_1$. Then either $(x_n)$ or $(y_n)$ contains a basic subsequence in $X$.
I'm sure this is discussed in some book out there somewhere, so a reference would be great. Thanks!