The present paper deals with integral classes $\xi_p\in H_{2p+1}(L_1^{2p+1}\times L_2^{2p+1})$ which are counterexamples for the Steenrod realization problem, where $L_1^{2p+1}$ is the $(2p+1)$-Lens space and $p\geq 3$ is a prime number. For $p=3$, this is the Thom's famous counterexample. We give a geometric description of this class using the theory of stratifolds. As a consequence, we obtain a geometric interpretation of the obstruction to realizability, in terms of the Atiyah-Hirzebruch spectral sequence.