Cold atomic gases have emerged as very versatile laboratory realizations for strongly correlated quantum systems in highly tunable setups. Computationally, it is desirable to study the systems directly in the continuum, as their natural description calls for. This is to avoid artifacts coming from lattice discretization, which affect the results of otherwise reliable methods like the density-matrix renormalization group (DMRG). In the last few years, work to reformulate these methods directly for low-dimensional quantum field theories (QFTs) using a continuum version of the quantum-information concept of matrix-product states (MPS) has been producing very encouraging results. I will provide a progress report of these efforts as seen from the fermionic front line. Interacting imbalanced Fermi gases constitute a nice playground for theories and experiments alike. Coincidentally, their physics is particularly rich in one dimension, where in the attractive case they can display algebraic superfluid order with an unusual pair-density wave character that is not found in higher dimensions. This is the one-dimensional analog of the elusive Fulde-Ferrell Larkin-Ovchinnikov (FFLO) pairing mechanism. Computational approaches are extremely valuable to study these systems beyond what rigorous analytic approaches can offer and thus make better contact with the experimental scenarios. In turn, this kind of theory-experiment dialogue is ideally suited to inform and validate the development of continuum matrix-product states (cMPS) as a specially tailored numerical framework for the optimally compressed description of low-dimensional QFTs.