Since some viruses share transmission routes, coinfection can occur. While most models assume one target cell, many viruses infect and replicate in multiple cell types. The purpose of this study is to develop and analyze a model describing coinfection by two viruses that grow and compete within two distinct target-cell populations. We prove that the proposed model is mathematically well-defined and admits unique, biologically meaningful solutions. Using the next-generation matrix method, we derive expressions for the basic reproduction numbers corresponding to virus type 1 single infection ( $R_1$), virus type 2 single infection ( $R_2$), and two-virus coinfection ( $R_0$). The model usually admits an infection-free equilibrium. The existence conditions for the virus type 1 single-infection, virus type 2 single-infection, and two-virus coexistence equilibria are also established. Applying the Lyapunov direct method, we demonstrate the global stability of all steady states. The obtained results reveal new insights into the factors that allow two viruses to coexist in a stable state, thereby enabling the possibility of chronic coinfections. The model is further extended to examine the influence of two reverse transcriptase (RT) inhibitors and to explore the role of a second target-cell population in two-virus codynamics. We find that neglecting the second target-cell population leads to underestimation of $R_1$ and $R_2$; consequently, drug levels determined from a one-target-cell model may be insufficient to clear the viruses. The model is extended to incorporate antiviral drug therapy and to determine the minimum drug efficacies required to eliminate viral coinfection. The results provide deeper insight into the dynamics of dual infections involving viruses that compete for distinct target-cell populations.