Reconstructing astrophysical and cosmological fields from observations is challenging. It requires accounting for non-linear transformations, mixing of spatial structure, and noise. In contrast, forward simulators that map fields to observations are readily available for many applications. We present a versatile Bayesian field reconstruction algorithm rooted in simulation-based inference and enhanced by autoregressive modeling. The proposed technique is applicable to generic (non-differentiable) forward simulators and allows sampling from the posterior for the underlying field. We show first promising results on a proof-of-concept application: the recovery of cosmological initial conditions from late-time density fields.