  We present a numerical study of competing orders in the 1D $t$-$J$ model with
long-range RKKY-like staggered spin interactions. By circumventing the
constraints imposed by Mermin-Wagner's theorem, this Hamiltonian can realize
long-range N\'eel order at half-filling. We determine the full phase diagram as
a function of the exchange and particle density using the density matrix
renormalization group (DMRG) method. We show that pairing is disfavored and the
AFM insulator and metallic phases are separated by a broad regime with phase
segregation, before spin-charge separation re-emerges at low densities. Upon
doping, interactions induce a confining potential that binds holons and spinons
into full fledged fermionic quasi-particles in a range of parameters and
densities. We numerically calculate the photoemission spectrum of the model,
showing the appearance of a coherent quasi-particle band splitting away from
the holon-spinon continuum with a width determined by $J$ that survives at
finite doping. Comparison with analytical results using the self-consistent
Born approximation (SCBA) and by solving the spinon-holon problem offer insight
into the internal structure of the quasi-particles and help us explain the
different features in the spectrum. We discuss how this simple toy-model can
teach us about the phenomenology of its higher dimensional counterpart.
