Generalized entanglement wedges (“holograms”) exist in arbitrary spacetimes. They exhibit suggestive properties such as strong subadditivity, nesting, and no-cloning.  But the entanglement wedges of AdS boundary regions satisfy an additional condition, complementarity: for a boundary subregion $B$ with boundary complement $\bar B$, minEW($B$) is the bulk complement of maxEW$(\bar B)$. 

Here we refine the definition of holograms and prove that they satisfy complementarity. We identify an analogue of $\bar B$, the region containing the fundamental degrees of freedom complementary to $B$: given a bulk wedge $a$, its \emph{fundamental complement} $\tilde a$ is the smallest wedge whose asymptotic boundary complements the past and future of $a$ on conformal infinity. We show that $\tilde a$ always lies inside the ordinary spacelike complement $a’$ of $a$. We then prove that the min-hologram $\emin(a)$ is the bulk complement,  in the spacetime $a’$, of the max-hologram $\emax(\tilde a)$.

Our refined prescription for holograms also remedies other shortcomings of the originial proposal. We exhibit several explicit examples of entanglement wedges and of fundamental complements, in asymptotically de Sitter, flat, and AdS spacetimes.