We analyse unstable D-brane systems in type-I string theory. Generalizing the proposal in hep-th/0108085, we give a physical interpretation for real KK-theory and claim that the D-branes embedded in a product space X x Y which are made from the unstable Dp-brane system wrapped on Y are classified by a real KK-theory group KKOp1 (X, Y). The field contents of the unstable D-brane systems are systematically described by a hidden Clifford algebra structure. We also investigate the matrix theory based on non-BPS D-instantons and show that the spectrum of D-branes in the theory is exactly what we expect in type-I string theory, including stable non-BPS D-branes with Z(2) charge. We explicitly construct the D-brane solutions in the framework of BSFT and analyse the physical property making use of the Clifford algebra.