  Motivated by recent trapped-ion quantum simulation experiments, we carry out
a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type
interactions that decay as $1/r^{\alpha}$, using a combination of finite and
infinite-size DMRG calculations, spin-wave analysis, and field theory. In the
absence of long-range interactions, varying the spin-coupling anisotropy leads
to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a
topological Haldane phase, and an antiferromagnetic Ising phase. If long-range
interactions are antiferromagnetic and thus frustrated, we find primarily a
quantitative change of the phase boundaries. On the other hand, ferromagnetic
(non-frustrated) long-range interactions qualitatively impact the entire phase
diagram. Importantly, for $\alpha\lesssim3$, long-range interactions destroy
the Haldane phase, break the conformal symmetry of the XY phase, give rise to a
new phase that spontaneously breaks a $U(1)$ continuous symmetry, and introduce
an exotic tricritical point with no direct parallel in short-range interacting
spin chains. We show that the main signatures of all five phases found could be
observed experimentally in the near future.
