Based on recent experimental and theoretical hints on possible formation of a resonant four-neutron system we study effects of appearance of such a cluster in neutron rich baryon matter inside neutron stars. For this purpose we employ a relativistic mean field approach which includes nucleons, Δ-baryons as well as light nuclear clusters. The Pauli blocking which suppresses tetraneutrons and stable clusters is explicitly included to the model as well. Our analysis demonstrates that tetraneutrons are able to exist as the Bose-Einstein condensate. Such a condensate weakens the nucleon Cooper pairing and significantly suppresses formation of the superfluid phase inside neutron stars. Tetraneutrons are also found to strongly feed vector meson fields and suppress Δ-baryons leading to stiffening of equation of state.