We study a system of strongly interacting one-dimensional (1D) bosons on a ring pierced by a synthetic magnetic flux tube. By the Fermi-Bose mapping, this system is related to the system of spin-polarized noninteracting electrons confined on a ring and pierced by a solenoid (magnetic flux tube). On the ring there is an external localized δ-function potential barrier, V (φ) = gδ(φ − φ0 ). We study the Berry phase associated with the adiabatic motion of the δ-function barrier around the ring as a function of the strength of the potential g and the number of particles N. The behavior of the Berry phase can be explained via quantum mechanical reflection and tunneling through the moving barrier which pushes the particles around the ring. The barrier produces a cusp in the density to which one can associate a missing charge q (missing density) for the case of electrons (bosons, respectively). We show that the Berry phase (i.e., the Aharonov-Bohm phase) cannot be identified with the quantity q/h¯ A · dl. This means that the missing charge cannot be identified as a (quasi)hole. We point out the connection of this result and recent studies of synthetic anyons in noninteracting systems. In addition, for bosons we study the weakly interacting regime, which is related to the strongly interacting electrons via Fermi-Bose duality in 1D systems.