I read somewhere that "it is easy" to construct a computable presentation for the model ($\omega$, <) so that any computable isomorphism between this construction and the usual presentation of $\mathbb{N}$ (with the usual order) would compute the halting set. So this would show that ($\omega$, <) is not computably categorical (clearly), but they didn't show the construction and I can't figure it out!
Does anyone know the construction?