This is in the Geometric Set Theory book with Jindra, in particular on pages 190-191 of
the version here: https://people.clas.ufl.edu/zapletal/files/balanced14.pdf
The partial order there forces over a Solovay model. The conditions are disjoint pairs of set of reals $(a,b)$, where $a$ is finite and $b$ is countable, with the order of coordinatewise inclusion.
The GST machinery shows that the partial order doesn't add reals. The union of the finite parts of a generic filter will then be a set of reals of cardinality less than the continuum such that, by genericity, every real is a sum of two of them. In fact, the set will be Dedekind-finite.