Multiphase compressible flows with surface tension and viscous effects arise in a wide range of applications, including liquid jet breakup in high-speed combustion chambers, droplet impact in supersonic atmospheric flight, and shock-induced interfacial instabilities. Accurate numerical simulation of such flows remains challenging due to complex interfacial physics and the presence of strong contact discontinuities. The Finite Volume method based on Exact two-phase Riemann problems (FIVER) framework was originally developed to address the numerical difficulties associated with strong interfacial discontinuities. In this work, the FIVER framework is extended to incorporate surface tension and viscous effects at fluid interfaces. Surface tension is embedded directly into the exact two-phase Riemann problem, thereby accounting for interfacial forces in a manner fully consistent with the underlying FIVER formulation. Viscous interfacial effects are treated using a selective ghost-node population strategy that enforces the appropriate interfacial jump conditions. In addition, solution strategies are proposed to accelerate the treatment of implicit time-integration methods for the class of problems considered. The extended framework is validated using standard benchmark problems, including the stationary droplet test for surface tension and the Richtmyer–Meshkov instability for viscous interfacial effects. All examples are computed using adaptive mesh refinement to efficiently resolve interfacial features and localized flow structures. The results demonstrate accurate resolution of interfacial dynamics and faithful representation of the underlying physics.