Systems of correlated quantum matter can be a steep challenge to any would-be method of solution. Matrix-product state (MPS)-based methods can describe 1D systems quasiexactly, but often struggle to retain sufficient bipartite entanglement to accurately approximate 2D systems already. Conversely, Quantum Monte Carlo (QMC) approaches, based on sampling a probability distribution, can generally approximate 2D and 3D systems with an error that decays systematically with growing sampling size. However, QMC can suffer from the so-called sign problem, that makes the approach prohibitively costly for many systems of interest, such as repulsively interacting fermions away from commensurate densities and frustrated systems. In this article, we introduce a new hybrid approach, that combines auxiliary-field QMC (AFQMC) with MPS-based algorithms. This hybrid technique removes or reduces the sign problem (depending on the specific model) while also needing to retain much lower bipartite entanglement than brute-force application of a MPS-algorithm, without the use of uncontrolled approximations. We present two use-cases of the algorithm that would be challenging or impossible to address with any other approach, and quantify the extent of any remaining sign problem.