  The spin-$\frac{1}{2}$ XXZ chain with easy-plane anisotropy in a transverse
field describes well the thermodynamic properties of the material ${\rm
Cs_2CoCl_4}$ in a wide range of temperatures and fields including the region
close to the spin-flop Ising quantum phase transition. For a comparison with
prospective inelastic neutron scattering experiments on this compound, we
present results of an extensive numerical study of its dynamic structure factor
$\mathcal{S}^{\alpha \beta}(k,\omega)$ using matrix-product-state (MPS)
techniques. Close to criticality, the dynamic part of the correlator
$\mathcal{S}^{xx}$ longitudinal to the applied field is incoherent and
possesses a small total weight as the ground state is already close to
saturation. The transverse correlator $\mathcal{S}^{zz}$, on the other hand, is
dominated by a coherent single-particle excitation with additional spectral
weight at higher energies that we tentatively attribute to a repulsively bound
pair of particles. With increasing temperature, the latter quickly fades and
spectral weight instead accumulates close to zero wavevector just above the
single-particle energy. On a technical level, we compare the numerical
efficiency of real-time evolution to an MPS-based Chebyshev expansion in the
present context, finding that both methods yield results of similar quality at
comparable numerical costs.
