We present numerical evidence for the presence of bound entanglement, in addition to distillable entanglement, between disjoint regions of the one-dimensional non-interacting scalar field vacuum. To reveal the entanglement structure, we construct a local unitary operation that transforms the high-body entanglement of latticized field regions into a tensor-product core of mixed (1 x 1) pairs exhibiting an exponential negativity hierarchy and a separable halo with non-zero entanglement. This separability-obscured entanglement (SOE) is driven by non-simultaneous separability within the full mixed state, rendering unentangled descriptions of the halo incompatible with a classically connected core. We quantify the halo SOE and find it to mirror the full negativity as a function of region separation, and conjecture that SOE provides a physical framework encompassing bound entanglement. Similar entanglement structures are expected to persist in higher dimension and in more complex theories relevant to high-energy and nuclear physics, and may provide a language for describing the dynamics of information in transitioning from quarks and gluons to hadrons.