A reconfigurable pinching-antenna system (PASS) is presented, endowing pinching antennas (PAs) with both amplitude- and phase-controllable radiation beyond conventional implementations. To characterize this feature, a general and physically consistent model is established for PASS via multiport network theory. Within this model, the fundamental constraint of ideal reconfigurability of PAs is identified, allowing the full control of signal amplitudes and phases. A practical directional-coupler (DC)-based PA model is then proposed, enabling both amplitude-only control and amplitude-constrained phase control. Beamforming optimization is investigated for both ideal and practical cases: an optimal solution is obtained for ideal PAs, whereas a high-quality iterative algorithm is developed for DC-based PAs. Numerical results suggest that in single-user scenarios: (i) with optimized PA positions, performance gains arise primarily from amplitude reconfigurability and DC-based PAs approach ideal performance, and (ii) with fixed PA positions, both amplitude and phase reconfigurability are critical and DC-based PAs incur non-negligible loss.