Step 1: Understanding water-gas shift reaction.
The reaction CO + H$_2$O → CO$_2$ + H$_2$ is used to produce hydrogen. When catalyzed homogeneously, the catalyst must dissolve in the reaction medium. Step 2: Identify the homogeneous catalyst.
Rhodium-phosphine complexes such as [RhCl(PPh$_3$)$_3$] are well-known homogeneous catalysts for CO–H$_2$O transformations including the water-gas shift reaction. Step 3: Elimination of other options.
Cr$_2$O$_3$ is heterogeneous.
PdCl$_2$ and Ru complexes do not function as standard homogeneous catalysts in this specific reaction. Step 4: Conclusion.
Thus, the homogeneous catalyst is [RhCl(PPh$_3$)$_3$].