  This paper concerns a diffuse interface model for the flow of two
incompressible viscoelastic fluids in a bounded domain. More specifically, the
fluids are assumed to be macroscopically immiscible, but with a small
transition region, where the two components are partially mixed. Considering
the elasticity of both components, one ends up with a coupled
Oldroyd-B/Cahn--Hilliard type system, which describes the behavior of two-phase
viscoelastic fluids. We prove the existence of weak solutions to the system in
two dimensions for general (unmatched) mass densities, variable viscosities,
different shear moduli, and a class of physically relevant and singular free
energy densities that guarantee that the order parameter stays in the
physically reasonable interval. The proof relies on a combination of a novel
regularization of the original system and a new hybrid implicit time
discretization for the regularized system together with the analysis of an
Oldroyd-B type equation.
