When twisted to angles near 1°, graphene multilayers provide a window on electron correlation physics. Here, we report the discovery of a correlated electron-hole state in double-bilayer graphene twisted to 2.37°. At this angle, the moiré states retain much of their isolated bilayer character, allowing their bilayer projections to be separately controlled by gates. We use this property to generate an energetic overlap between narrow isolated electron and hole bands with good nesting properties. Our measurements reveal the formation of ordered states with reconstructed Fermi surfaces, consistent with a density-wave state. This state can be tuned without introducing chemical dopants, enabling studies of correlated electron-hole states and their interplay with superconductivity.