  We prove that if $\bI$ is a p.\ o. set in a countable transitive model $\gM$
of $\ZFC$ then $\gM$ can be extended by a generic sequence of reals $\a_\i,$
$\i\in\bI,$ such that $\aleph_1^\gM$ is preserved and every $\a_\i$ is Sacks
generic over $\gM[\ang{\a_\j:\j<\i}]$.
  The structure of the degrees of \dd\gM constructibility of reals in the
extension is investigated.
  As an application, we obtain a model in which the $\is12$ equivalence
relation $x \E y$ iff $\rL[x]=\rL[y]$ ($x,\,y$ are reals) does not admit a
reasonable form of the Glimm -- Effros theorem.
