The finite element method (FEM) and its associated field have mainly been developed for adiabatic and closed systems. Nonetheless, open systems, which allow for the exchange of energy and mass with the surroundings, have gained increasing interest in applications where mass change occurs. For solving open systems two approaches can be undertaken. The first is the local approach, which incorporates mass change as an internal variable at the material level, while the second is the global approach, which treats mass change as an additional degree of freedom (DOF), solving the mass and momentum balance equations simultaneously. Although the global approach has been already developed, it has not yet incorporated a kinematic split of the deformation gradient. This split is necessary for modeling large strain deformations volume change (e.g. soft tissues). Hence, this study proposes a monolithic coupled mass-mechanical framework with a multiplicative split of the deformation gradient. The deformation gradient is multiplicatively split into mass-changing and mechanical components, with the mass-changing part accommodating orthotropic deformation and constraints enforcing density preservation. The study presents the complete finite element method from the kinematic foundations through to the discretization process. A sensitivity analysis is conducted to study the effects of various factors on the deformation and mass change. Moreover, a numerical example demonstrating the framework's application to a general mass change problem is also conducted. The results show that the proposed framework effectively models mass-changing phenomena, offering a tool for future research in the field of open systems.