Type-II quantum well heterostructures are considered high-potential candidates for next-generation active semiconductor devices. They promise low fundamental transition energies and suppressed Auger scattering as well as temperature stability in device performance. Understanding their fundamental properties, such as scattering and diffusion, and revealing intricate differences from type-I quantum structures are important steps towards optimized structure design. Using degenerate four-wave mixing spectroscopy, we investigate the phase coherence of excitonic polarizations in a (Ga,In)As/GaAs/Ga(As,Sb) type-II double-quantum-well structure. It is designed to exhibit spectrally isolated resonances in its linear absorption spectrum including a well-resolved charge-transfer exciton resonance. This allows us to study the coherent dynamics of the unperturbed charge-transfer exciton polarization. In addition, the effects of many-body interactions with free charge carriers as well as excitons that are injected by an optical prepulse are revealed. Scattering of charge-transfer excitons with free charge carriers is three times more efficient than scattering of charge-transfer excitons with each other. The comparison with Wannier excitons in a type-I quantum well structure shows that the interaction strength between charge-transfer excitons is about twice as large as for excitons in a type-I quantum well structure.