  The self-consistent matter coupling is found in a broad class of minimally
modified gravity theories which was discovered recently. All constraints in the
theories remain first class and thus a graviton has only 2 local degrees of
freedom. The cosmological solution of one of the examples in this class, the
so-called square root gravity, exhibits a singularity freeness at high energy
limit. At low energy limit, the theory smoothly connects to GR. A general
feature of the theories in this class, with the self-consistent matter coupling
discovered in our current work, is the non-trivial interaction among different
components of matter sector. We have also checked the Hamiltonian structure of
a scalar QED coupling to the square root gravity in the same manner. All
constraints in the theory are first class too and thus the local U(1) gauge
symmetry in scalar QED is preserved.
