Anyonic exchange statistics can emerge when the configuration space of quantum particles is not simply-connected. Most famously, anyon statistics arises for particles with hard-core two-body constraints in two dimensions. Here, the exchange paths described by the braid group are associated to non-trivial geometric phases, giving rise to abelian braid anyons. Hard-core three-body constraints in one dimension (1D) also make the configuration space of particles non-simply connected, and it was recently shown that this allows for a different form of anyons with statistics given by the traid group instead of the braid group. In this article we propose a first concrete model for such traid anyons. We first construct a bosonic lattice model with number-dependent Peierls phases which implement the desired geometric phases associated with abelian representations of the traid group and then define anyonic operators so that the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model shows various indications of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum traid-anyonic wave functions. This provides not only an a-posteriori justification of our model, but also shows that our construction serves as an intuitive approach to traid anyons. Moreover, it contrasts with the non-Galilean invariant continuum limit of the anyon-Hubbard model [Keilmann et al., Nat. Comm. 2, 361 (2011)] describing braid anyons on a discrete 1D configuration space. We attribute this difference to the fact that (unlike braid anyons) traid anyons are well defined also in the continuum in 1D.