The product of Eilenberg-MacLane spaces $K(\mathbb{Z}/2,1) \times K(\mathbb{Z}/2,2)$ is an $H$-space with respect to a "twisted product structure", where you add the product of the first entries to the second, i.e. $$ (a,b)\circ (a', b') := (a + a', b+b'+a\cdot a'), $$ where the $+$ denotes the additive $H$-space structure on $K(\mathbb{Z}/2,i)$ and $\cdot \colon K(\mathbb{Z}/2,1) \times K(\mathbb{Z}/2,1) \to K(\mathbb{Z}/2,2)$ is induced by the multiplication in the Eilenberg-MacLane spectrum. This structure describes for example the addition of the (low-dimensional) twists of $KO$-theory.

The above turns the homology $H_*(K(\mathbb{Z}/2,1) \times K(\mathbb{Z}/2,2); \mathbb{Z}/2)$ into a ring. Inspection of the low degrees shows that the ring structure does not coincide with the "untwisted" one. Is there a nice description of this ring? Has it been studied in the literature?

Brown-Peterson homology: An introduction and sampler. (He does it for an odd prime, but the even case is actually dramatically simpler.) Just reading that will give you a compact description of the behavior in all degrees, and I suspect that anything you'd mean by "nice" could be teased out of that. $\endgroup$ – Eric Peterson Jul 19 '13 at 16:42