Numerical simulation of multiphase flow and transport processes forms an important base for the assessment of deep geological repositories for radioactive waste. The finite volume simulation code TOUGH2-MP with its EOS7R equation-of-state module is a good starting point for large-scale simulations of the relevant processes, including solute transport of radionuclides, in and around a geological repository. On this base, we developed the equation-of-state module EOS75Rx that contains optimizations and specific extensions allowing for a much more efficient treatment of the problem at hand. First, hydrogen, which is formed by corrosion of waste containers and by radiolysis of organic wastes, replaces air as the main component of the gas phase. Second, an arbitrary number of variably long decay chains with branching can be considered. Third, solubility limitation and associated precipitation of chemical elements are modeled. Finally, a bug fix related to the source terms has been implemented. The new TOUGH2-MP EOS75Rx module has been validated using unitary tests and benchmark problems for the single- and two-phase flow and transport of radionuclides through porous media and soils. Its performance has been demonstrated by a large-scale, three-dimensional simulation of the performance of a generic deep geological repository in clay host rock.