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With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.

Is there an explicit description of a cobordism between them?

I know the answer for $n = 1$: it reduces to finding a cobordism between $S^2$ and $S^1\times S^1$ (take a solid ball and remove a solid torus from the interior, or take a solid torus and remove a solid ball from the interior).

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An explicit cobordism is given by Stong:

R. E. Stong, A Cobordism, Proceedings of the American Mathematical Society Vol. 35, No. 2 (Oct. 1972), pp. 584-586

I do like the short title "A Cobordism".

The construction is as follows:

Write $\mathbb{CP}^n$ with homogeneous coordinates $[z_0,z_1,\dots ,z_n]$ and consider the space $$W=(\mathbb{CP}^n\times [0,1])/\sim,$$ where $([z_0,z_1,\dots ,z_n],1)\sim([\bar{z_0},\bar{z_1},\dots ,\bar{z_n}],1)$ if $|z_0^2+z_1^2\dots +z_n^2|\leq\frac{3}{5}$. Then Stong takes a page to prove that $W$ is the desired cobordism.

But one should also read the note: enter image description here enter image description here

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