  We describe the imaginary sorts of infinite products in terms of imaginary
sorts of the factors. We extend the result to certain reduced powers and then
to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the
ideal of finite subsets of $I$. As a special case, using the
Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the
imaginary sorts of the ring of rational adeles. Our methods include the use of
the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning
from monadic second order imaginaries to first-order reducts, and for proving a
certain ``one-way'' model-theoretic orthogonality within the adelic
imaginaries.
