  We construct a model $M$ of ZF which lies between $L$ and $L[c]$ for a Cohen
real $c$ and does not have the form $L(x)$ for any set $x$. This is loosely
based on the unwritten work done in a Bristol workshop about Woodin's HOD
Conjecture in 2011. The construction given here allows for a finer analysis of
the needed assumptions on the ground models, thus taking us one step closer to
understanding models of ZF, and the HOD Conjecture and its relatives. This
model also provides a positive answer to a question of Grigorieff about
intermediate models of ZF, and we use it to show the failure of Kinna--Wagner
Principles in ZF.
