Coalescence process of binary immiscible fluid mixtures having bimodal size distributions, prepared by mixing two pre-sheared samples at different shear rates, ${gamma}_{pre1};and;{gamma}_{pre2}$, under shear flow at a final shear rate, ${gamma}_f$, are examined by transient shear stress measurements and microscopic observations in comparison with the results for simply pre-sheared samples having narrow size distributions (unimodal distribution samples). Component fluids are a silicone oil (PDMS) and a hydrocarbon-formaldehyde resin (Genelite) and their viscosities are 14.1 and 21.0 $pa{cdot}sec$ at room temperature $(ca.;20^{circ}C)$, respectively. The weight ratio of PDMS: Genelite was 7:3. Three cases, $({gamma}_{pre1}=7.2sec^{-1},;{gamma}_{pre2}=12.0sec^{-1};and;{gamma}_f=2.4sec^{-1}),;({gamma}_{pre1}=0.8sec^{-1},;{gamma}_{pre2}=4.0sec^{-1};and;{gamma}_f=2.4sec^{-1}),;and;({gamma}_{pre1}=7.2sec^{-1},;{gamma}_{pre2}=12.0^sec^{-1};and;{gamma}_f=7.2sec^{-1})$ the first case, transient shear stress did not show any significant difference but domains larger than the initial state are observed at short times. In the latter cases, there exist undershoot of shear stress, reflecting existence of deformed large domains, which is confirmed by the direct observation. It is concluded that coalescence between large and small domains more frequently occur than coalescence between the domains with similar size in the bimodal distribution samples.