A new mass-conserving and quasi-monotonic antidiffusive filter has been designed for application in semi-Lagrangian-type models. The filter redistributes local mass such that the resulting values are brought closer to target values. The target values, in turn, are specified from a nonlinear antidiffusion of the original nonfiltered forecast. To achieve quasi-monotonicity, the target values are constrained to a certain interval, defined by the minimum and maximum value of the upstream grid cells surrounding the semi-Lagrangian departure point. Allowing a less local reorganization of mass the filter can be made fully monotonic and positive definite. The filter has been applied to the recently developed locally mass-conserving semi-Lagrangian (LMCSL) scheme. A number of idealized test simulations in one and two dimensions demonstrates that the combined scheme is stable and quasi monotonic. Furthermore, the accuracy is enhanced considerably as compared to the original LMCSL scheme, particularly near sharp changes in gradients. When tested in the geophysical flow environment of a shallow-water model, the proposed filter, in practice, ensures monotonicity and positive definiteness without generation of spurious features. In its present implementation the computational cost of the combined scheme is approximately twice the cost of the LMCSL scheme.