  We show that a separable real Banach space embeds almost isometrically in a
space $Y$ with a shrinking 1-unconditional basis if and only if $\lim_{n \to
\infty} \|x^* + x_n^*\| = \lim_{n \to \infty} \|x^* - x_n^*\|$ whenever $x^*
\in X^*$, $(x_n^*)$ is a weak$^*$-null sequence and both limits exist. If $X$
is reflexive then $Y$ can be assumed reflexive. These results provide the
isometric counterparts of recent work of Johnson and Zheng.
