Characteristic classes of vector bundles on a real algebraic variety

For a vector bundle $E$ on a real algebraic variety $X$, the author studies the connections between the characteristic classes
$$ c_k(E(\mathbf C))\in H^{2k}(X(\mathbf C),\mathbf Z),\quad w_k(E(\mathbf R))\in H^k(X(\mathbf R),\mathbf F_2). $$
It is proved that for $M$-varieties the equality $w_k(E(\mathbf R))=0$ implies the congruence $c_k(E(\mathbf C))\equiv 0 \operatorname{mod}2$. Sufficient conditions are found also for the converse to hold; this requires the construction of new characteristic classes $cw_k(E(\mathbf C))\in H^{2k}(X(\mathbf C);G,\mathbf z(k))$. The results are applied to study the topology of $X(\mathbf R)$.