Acoustic metamaterials allow for creating selective pass- and stop-bands on the frequency spectrum. We demonstrate the possibility of designing acoustic metamaterials as core-shell 2D-phononic media with an extremely simple morphology, the frequency spectrum of which contains many real-time tunable bandgaps. The connected shells of such metamaterials form a grid with square cells filled with nuclei partitionable into two subsystems. Both subsystems are characterized by their frequency spectra, and it is the coupling between them that generates the bandgaps. If the structural elements of the metamaterial are built based on magnetoelastomers, then bandgaps can be easily controlled by an external magnetic field that changes the elastic moduli of shells/cores. We have shown the possibility of manipulating single bandgaps in different parts of the spectrum, and simultaneous control of all bandgaps up to their complete disappearance. This manipulation can be carried out, specifically, with no change in the maximum achievable frequency in the metamaterial. The results obtained can be used for selective filtering of damaging wave components, active control of seismic or blast waves, sonar systems, ultrasound imaging, impact-resistant structures, and noise cancellation protocols. The physical concepts developed are extendable to 3D-structures in a similar fashion so can benefit a wider community.