We report on the growth of GaSb shells on Au-catalyzed GaAs or GaP nanowires (NWs) using metalorganic vapor phase epitaxy. The large lattice mismatch between GaSb and GaAs (GaP), 7.8% (11.8%), results in surface roughening and GaSb island formation via the Stranski-Krastanov (S-K) growth mode. Based on transmission electron microscopy (TEM) analysis, coherent GaSb islands on GaAs NWs could be grown up to a thickness of 1.8 nm for a core diameter of 34 ± 5 nm. For greater shell thickness of 9 ± 3 nm, equal axial and radial strain relaxation occurred increasing from 74% ± 3% for GaAs/GaSb NWs and 91% ± 2% for GaP/GaSb NWs to 100% with increasing core diameter from 15 ± 2 nm to 55 ± 3 nm. Axial strain is relieved by periodic misfit dislocations with edge components parallel to the growth direction. Tangential relaxation is presumed to occur partially by roughening via the S-K growth mode but dislocations with edge components perpendicular to the growth direction were not detected. Raman scattering measurements were performed on ensembles of NWs and the absolute residual strain in the core and shell were determined from the shift of the zone-center phonon modes. Raman results were consistent with the TEM analysis. It was found that the residual strain is higher in GaAs/GaSb NWs (7.3%) compared to GaP/GaSb NWs (1.7%).