  We compute the phase diagram of a one-dimensional model of spinless fermions
with pair-hopping and nearest-neighbor interaction, first introduced by Ruhman
and Altman, using the density-matrix renormalization group combined with
various analytical approaches. Although the main phases are a Luttinger liquid
of fermions and a Luttinger liquid of pairs, we also find remarkable phases in
which only a fraction of the fermions are paired. In such case, two situations
arise: either fermions and pairs coexist spatially in a two-fluid mixture, or
they are spatially segregated leading to phase separation. These results are
supported by several analytical models that describe in an accurate way various
relevant cuts of the phase diagram. Last, we identify relevant microscopic
observables that capture the presence of these two fluids: while originally
introduced in a phenomenological way, they support a wider application of
two-fluid models for describing pairing phenomena.
