Let us first watch the time-evolution of a free relativistic cat-state:
Now, let us observe the evolution of the corresponding particle-antiparticle coherent superposition:
In this case we observe that the antiparticle at the bottom advances to the positive direction despite of having a negative momentum. This type of dynamics is characteristic of antiparticles!
One critical piece of information is that both states are undergoing the same degree of quantum decoherence due to the contact with an environment. Nevertheless, the particle-antiparticle superposition maintains the interference robust. We just found a state that belongs to a free-decoherence space!
Al should also mention that the relativistic state happens to be a Majorana state.
 Dirac open-quantum-system dynamics: Formulations and simulations,
Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz Phys. Rev. A 94, 052111 – Published 14 November 2016