The need for suitable many- or infinite-fermion correlation functions to describe strongly correlated systems is discussed, and the question linked to the need for a correlated basis, in which the ground state may be positive definite for certain low-dimensional geometries. In seeking a positive-definite basis a particular trial basis is proposed, based on that for hard-core bosons in pure one-dimensional systems. Single-particle correlations in this basis are evaluated for the case of the ground state of a quasi-1D Hubbard model in the limit of extreme correlation. The model is a strip of the 2D square lattice wrapped around a cylinder, and is related to a ladder geometry with periodic boundary conditions along its edges. This is done using both a novel mean-field theory and exact diagonalization, and the basis is indeed found to be well suited for examining (quasi-) order in the model. The model has a paramagnetic region and a Nagaoka ferromagnetic region. In the numerical calculation the correlation function in the paramagnetic phase has a power-law decay and the charge motion is qualitatively hard-core bosonic. The mean field leads to an example of a BCS-type model with single-particle bosonic long-range order.