A series of poly(ethylene-co-acrylic acid)-cb-atactic poly-propylene (EAA-cb-aPP) comb block copolymers were synthesized by grafting aPP-OH macromonomers onto a commercial EAA copolymer made by the high-pressure free radical process. The starting EAA copolymer contains 11 wt % of EAA units and has a significant amount of long chain branches. Therefore, the EAA-cb-aPP copolymers can be classified as hyper branched. Room temperature atomic force microscopy and X-ray scattering measurements reveal strong, finely textured, phase segregation of the amorphous aPP and semicrystalline EAA domains, which persists in the melt state. The amorphous aPP side chains have an unexpected nucleating effect that facilitates crystallization of the EAA backbone, as evidenced by an increase in crystallization temperature. Moreover, phase segregation has a strong effect on both the linear and nonlinear viscoelastic response of the copolymers. Increases in both the branching density and branch chain length result in an improvement of melt strength as well as an increase in the extensional strain hardening (SH). We postulate that the SH enhancement may arise from the interfacial anchoring of the aPP side chains in the aPP homopolymer domains. This would produce additional resistance for the EAA backbone to stretch under uniaxial load due to an energetically unfavorable process of pulling the aPP arms into the EAA phase where they would face strong repulsions.