Let us not forget about Pilot-Wave theory, where it has both particle and wave-like properties simultaneously. In fact, I don't quite get why people are so enamored by these fanciful interpretations when Pilot-Wave is so much more down-to-earth.
The physical, conceptual differences between any quantities describing droplets on one side and the wave function on the other side are clear. The former are observable – you may actually measure what the shape of the droplet looks like; you can't measure the wave function by any apparatus, at least not in a single repetition of the experiment. The former has an objective interpretation; the latter has a probabilistic interpretation, and so on. The wave function just encodes all the probability distributions for actual observables – but the wave function isn't and can't be one of them.
http://motls.blogspot.com/2014/07/droplets-and-pilot-waves-v...
http://iopscience.iop.org/1742-6596/306/1/012047/pdf/1742-65...
I thought the difficulty with pilot-wave theory was that it was difficult, perhaps impossible, to reconcile with special relativity.