Decoherence is the process via which quantum superposition states are reduced to classical mixtures. Decoherence has been predicted for relativistically accelerated quantum systems; however, examples to date have involved restricting the detected field modes to particular regions of space-time. If the global state over all space-time is measured, then unitarity returns and the decoherence is removed. Here, we study a decoherence effect associated with accelerated systems that cannot be explained in this way. In particular, we study a uniformly accelerated source of a quantum field state—a single-mode squeezer. Even though the initial state of the field is vacuum (a pure state) and the interaction with the quantum source in the accelerated frame is unitary, we find that the final state detected by inertial observers appears to be decohered, i.e., in a mixed state. This unexpected result may indicate new directions in resolving inconsistencies between relativity and quantum theory. We extend this result to a two-mode state and find that entanglement is also decohered.