  Over the real numbers with $\Z/2-$coefficients, we compute the
$C_2$-equivariant Borel motivic cohomology ring, the Bredon motivic cohomology
groups and prove that the Bredon motivic cohomology ring of the real numbers is
a proper subring in the $RO(C_2\times C_2)$-graded Bredon cohomology ring of a
point.
  This generalizes Voevodsky's computation of the motivic cohomology ring of
the real numbers to the $C_2$-equivariant setting. These computations are
extended afterwards to any real closed field.
