Are there Alexander-Whitney maps in geometric homology?

When $X$ is a smooth manifold, Lipyanskiy defines a chain complex whose homology is isomorphic to singular homology - let's say $GC_*(X)$ - generated by maps $\sigma: M \to X$ from compact $k$-manifolds with corners $M$. Working over $\Bbb F_2$ to avoid orientation discussions, $GC_k(X)$ is the free vector space generated by isomorphism classes of smooth maps $M \to X$ with connected domain modulo the subspace of "degenerate chains" (a chain has small image if its image can be covered by the image of a smooth manifold of strictly smaller dimension; a chain is degenerate if $\sigma$ and $\partial \sigma$ have small image). This is useful so that one can easily define intersection and fiber product maps on the chain level.

There is of course a natural product map $GC_*(X) \otimes GC_*(Y) \to GC_*(X \times Y)$ which is a homology isomorphism given by multiplying the chains on the nose. I would like for there to be an associative natural transformation $AW: GC_*(X \times Y) \to GC_*(X) \otimes GC_*(Y)$ giving a homotopy inverse to the product map. (Ideally, it would be a right inverse to the product map on the nose, but I'm not convinced this is possible.) Is there such a transformation? This is certainly true in other non-simplicial settings (in cubical homology, for instance), and it seems plausible.

If there's not a good choice of such, are there mild alterations I can make to the definition of $GC$ so that there is? (In particular, I would like $GC_*(G)$ to naturally be a bialgebra when $G$ is a Lie group.)