  In Mathias forcing, conditions are pairs $(D,S)$ of sets of natural numbers,
in which $D$ is finite, $S$ is infinite, and $\max D < \min S$. The Turing
degrees and computational characteristics of generics for this forcing in the
special (but important) case where the infinite sets $S$ are computable were
thoroughly explored by Cholak, Dzhafarov, Hirst, and Slaman~\cite{CDHS-2014}.
In this paper, we undertake a similar investigation for the case where the sets
$S$ are members of general countable Turing ideals, and give conditions under
which generics for Mathias forcing over one ideal compute generics for Mathias
forcing over another. It turns out that if $\mathcal{I}$ does not contain only
the computable sets, then non-trivial information can be encoded into the
generics for Mathias forcing over $\mathcal{I}$. We give a classification of
this information in terms of computability-theoretic properties of the ideal,
using coding techniques that also yield new results about introreducibility. In
particular, we extend a result of Slaman and Groszek and show that there is an
infinite $\Delta^0_3$ set with no introreducible subset of the same degree.
