The aim of this study was to investigate the structure–properties relationship of ternary blends of polylactide/polyhydroxybutyrate (PLA/PHB)/organo-modified layered silicate (OMLS). Morphological, thermal, rheological, and mechanical characterizations were performed to understand the influence of OMLS on PLA/PHB (70/30 wt%) formulations optimized through modifications with an epoxy-based chain extender, the use of a plasticizer, as well as the influence of the type of processing route: injection molding or fused filament fabrication. The addition of OMLS allowed the blend compatibility to be improved, with the appearance of a single melting peak on DSC thermograms at 146 °C, as well as the reduction in the size of the nodules for the injected molded specimens. Concerning the printed samples, AFM analysis revealed a coalescence of the PHB minor phase due to its degradation. This phenomenon was dramatically enhanced in the presence of OMLS and has been ascribed to the degradation of both the organo-modifier and the PHB minor phase in the blend. Rheological and mechanical tests (17% decrease in Young’s modulus and 13% decrease in elongation at break) confirmed this degradation that would have occurred during the manufacturing of the filaments and the printing of specimens due to additional thermal and cooling steps.