We investigate the phase behavior of model ternary triacylglycerol blends, comprising triolein (C57H104O6, OOO), tripalmitin (C51H98O6, PPP) and tristearin (C57H110O6, SSS), building upon extensive characterisation of single and binary mixtures, in order to rigorously map the thermal transitions of model natural 'fats'. A combination of calorimetry, X-ray diffraction, and FTIR spectroscopy is employed to determine crystallisation and melting temperatures and identify the corresponding phases in the complex ternary system. We recover the eutectic behaviour of SSS-PPP blends and the invariability of OOO neat transitions, and resolve the complex β' + β ternary surface, reflecting the roles of unsaturation and polymorphism of its constituents. Our results provide a representation of the OOO:PPP:SSS:temperature phase behaviour into a triangular prism, consistent with binary pair-wise data, which can inform a range of food science, cosmetic, pharmaceutical and cleaning applications that depend strongly on the physical-chemistry of such multicomponent 'triglycerides'.