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The emergent fluctuating hydrodynamics of a viscoelastic fluid modeled by the multiparticle collision dynamics (MPC) approach is studied. The fluid is composed of flexible, Gaussian phantom polymers, which interact by local momentum-conserving stochastic MPC collisions. For comparison, the analytical solution of the linearized Navier-Stokes equation is calculated, where viscoelasticity is taken into account by a time-dependent shear relaxation modulus. The fluid properties are characterized by the transverse velocity autocorrelation function in Fourier space as well as in real space. Various polymer lengths are considered---from dumbbells to (near-)continuous polymers. Viscoelasticity affects the fluid properties and leads to strong correlations, which overall decay exponentially in Fourier space. In real space, the center-of-mass velocity autocorrelation function of individual polymers exhibits a long-time tail independent of polymer length, which decays as $t^{-3/2}$, similar to a Newtonian fluid, in the asymptotic limit $t \to \infty$. Moreover, for long polymers an additional power-law decay appears at time scales shorter than the longest polymer relaxation time with the same time dependence, but negative correlations, and the polymer length dependence $L^{-1/2}$. Good agreement is found between the analytical and simulation results.