Suppose we have a ctm $M$ and $x, y$ Laver generic reals over $M$ so that $M[x] = M[y]$ (recall that Laver forcing is minimal, so that if $x \in M[y]$ then we already have $M[x] = M[y]$). Is there any "nice" relation between $x$ and $y$?
In many ways Laver forcing is close to Mathias forcing, so one particular kind of relation between $x$ and $y$ which I had in mind (but I am also interested in other kinds of relations) was the following fact which holds for $x, y$ Mathias generics:
If $z$ is a Mathias generic over a ctm $M$ and $x, y \subseteq z$ then $M[x] = M[y]$ iff $x E_0 y$, where $x E_0 y$ iff $x \Delta y$ is finite. (This is Claim 8.20 from Canonical Ramsey theory on Polish spaces by Kanovei, Sabok and Zapletal)
Since I think that minimality implies that there cannot be anything as nice for Laver forcing as there is for Mathias, here is the follow up question:
We know that we can get a Laver tree $p$ of Laver generics over $M$ (since $M$ is countable, see The Kunen-Miller chart by Judah and Shelah). Is there a Laver tree $q$ with $q \leq_0 p$ (meaning that $q$ has the same stem as $p$) where there is some "nice" relation between its branches? (or just $q \leq p$?)