Aqueous two-phase systems and water-in-water emulsions have attracted much recent interest. Here, we theoretically study the interactions of such systems with biomimetic membranes and giant unilamellar vesicles (GUVs). For partial wetting, the water-water interface and the membrane form a three-phase contact line that partitions the membrane into two distinct segments with different tensions and different curvature-elastic properties. On the nanometer scale, the capillary forces arising from the water-water interface lead to a smoothly curved membrane that forms an intrinsic contact angle with the interface. The corresponding balance conditions are derived here for general curvature-elastic parameters of the two membrane segments. On the micrometer scale, the capillary forces deform the membrane segments into spherical caps with an apparent kink along the contact line. A new computational method is introduced by which these piece-wise spherical vesicle shapes can be analyzed in a systematic manner. The method is based on a general relationship that is reminiscent of Neumann's triangle but depends explicitly on the curvatures of the membrane segments. For certain regions of the parameter space, corresponding to small or large spontaneous curvatures, the force balance along the apparent contact line can be described in a self-consistent manner and then leads to curvature-independent relationships that involve the total membrane tensions. The different relationships can be used to determine the material parameters of the droplet-vesicle system from the observed morphologies of the GUVs. The approach described here is quite general and can be applied to different membrane compositions and aqueous two-phase systems. The same computational approach can also be used to elucidate the response of biological membranes to the recently discovered membrane-less, droplet-like organelles.