  For uncountable downwards closed subtrees $U$ and $W$ of an $\omega_1$-tree
$T$, we say that $U$ and $W$ are strongly almost disjoint if their intersection
is a finite union of countable chains. The tree $T$ is strongly non-saturated
if there exists a strongly almost disjoint family of $\omega_2$-many
uncountable downwards closed subtrees of $T$. In this article we construct a
c.c.c. forcing which adds a Suslin tree together with a family of
$\omega_2$-many strongly almost disjoint automorphisms of it (and thus the
square of the Suslin tree is strongly non-saturated). To achieve this goal, we
introduce a new idea called $\rho$-separation, which is an adaptation to the
finite context of the notion of separation which was recently introduced by
Stejskalov\'{a} and the first author for the purpose of adding automorphisms of
a tree with a forcing with countable conditions.
