We investigate the role of Bose-Einstein condensation (BEC) of anti-kaons on the equation of state (EoS) and other properties of compact stars. In the framework of relativistic mean field model we determine the EoS for β-stable hyperon matter and compare it to the situation when anti-kaons condense in the system. We observe that anti-kaon condensates soften the EoS, thereby lowering the maximum mass of the stars. We also demonstrate that the presence of antikaon condensates in the high density core of compact stars may lead to a new mass sequence beyond white dwarf and neutron stars. The limiting mass of the new sequence stars is nearly equal to that of neutron star branch though they have distinctly different radii and compositions. They are called neutron star twins.