Multistate quantum logic gates provide a promising approach to increasing the dimensionality of information encoding and enhancing the flexibility of optical-field manipulation. In this work, we propose a scheme for realizing tri-state Pauli-type logic gates through the superposition of composite solitons in a cold-atom medium. An inverted-Ψ-type five-level cold atomic system is considered, and three coupled nonlinear Schrödinger (NLS) equations are derived from the Maxwell-Bloch equations by means of a singular perturbation expansion. The components of the exact composite-soliton solutions of the coupled NLS equations are employed to encode a three-state logical state. By introducing appropriate rotation operations, tri-state Pauli-X, Pauli-Y, and Pauli-Z operations are constructed. A dark-bright-bright threecomponent composite soliton is further used as an example to investigate the free propagation of the encoded state and its evolution under the gate operations. The results show that the Pauli-X gate exchanges the first and third components, the Pauli-Y gate performs component exchange accompanied by phase modulation, whereas the Pauli-Z gate produces a relative phase flip. These results demonstrate a feasible scheme for implementing tri-state logic operations with three-component composite solitons and provide a theoretical framework for multistate logic gates and multichannel optical information processing.