Diffuse interface methods have proven their ability to simulate complex two-phase flows. A number of robust numerical schemes have been developed to simulate such flows involving large density and pressure ratios. Diffusion induced by these methods, however, makes it difficult to localize the interface between the two fluids. To overcome this issue, while retaining the advantages of diffuse interface methods, a second-order extension using a Monotonic Upstream-centered Scheme for Conservation Laws-type (MUSCL-type) method of the implicit-explicit acoustic-transport splitting scheme introduced in [40] is presented. A specific compressive limiter is used for the volume fraction in order to limit the diffusion of the interface between the two fluids. Numerical simulations are presented to illustrate the capability of the proposed new method to simulate highly complex compressible two-phase flows.