  Let $X$ be a compact connected K\"ahler manifold equipped with an
anti-holomorphic involution which is compatible with the K\"ahler structure.
Let $G$ be a connected complex reductive affine algebraic group equipped with a
real form $\sigma_G$. We define pseudo-real principal $G$--bundles on $X$;
these are generalizations of real algebraic principal $G$--bundles over a real
algebraic variety. Next we define stable, semistable and polystable pseudo-real
principal $G$--bundles. Their relationships with the usual stable, semistable
and polystable principal $G$--bundles are investigated. We then prove that the
following Donaldson--Uhlenbeck--Yau type correspondence holds: a pseudo-real
principal $G$--bundle admits a compatible Einstein-Hermitian connection if and
only if it is polystable. A bijection between the following two sets is
established: 1) The isomorphism classes of polystable pseudo-real principal
$G$--bundles such that all the rational characteristic classes of the
underlying topological principal $G$--bundle vanish. 2) The equivalence classes
of twisted representations of the extended fundamental group of $X$ in a
$\sigma_G$--invariant maximal compact subgroup of $G$. (The twisted
representations are defined using the central element in the definition of a
pseudo-real principal $G$--bundle.) All these results are also generalized to
the pseudo-real Higgs $G$--bundle.
