The combination of core/shell geometry and band gap engineering in nanowire heterostructures can be employed to realize systems with novel transport and optical properties. Here, we report on the growth of InAs/InP/GaAsSb core-dual-shell nanowires by catalyst-free chemical beam epitaxy on Si(111) substrates. Detailed morphological, structural, and compositional analyses of the nanowires as a function of growth parameters were carried out by scanning and transmission electron microscopy and by energy-dispersive X-ray spectroscopy. Furthermore, by combining the scanning transmission electron microscopy-Moire technique with geometric phase analysis, we studied the residual strain and the relaxation mechanisms in this system. We found that InP shell facets are well-developed along all the crystallographic directions only when the nominal thickness is above 1 nm, suggesting an island-growth mode. Moreover, the crystallographic analysis indicates that both InP and GaAsSb shells grow almost coherently to the InAs core along the 112 direction and elastically compressed along the 110 direction. For InP shell thickness above 8 nm, some dislocations and roughening occur at the interfaces. This study provides useful general guidelines for the fabrication of high-quality devices based on these core-dual-shell nanowires.