There is a technique in homotopy theory called 'Zabrodsky mixing'. One can construct, for example, a finite CW-complex $X$ with the following properties:
1. $X$ is an $H$-space, i.e. it has a multiplication map $X\times X \to X$, which is associative and unital up to homotopy.
2. The $2$-localization of $X$ is equivalent to the $2$-localization of the Lie group $Sp(2)$ (as an $H$-space).
3. The $3$-localization of $X$ is equivalent to the $3$-localization of $S^3\times S^7$ (as an $H$-space).
This is just one example of the general procedure of piecing well-known $H$-spaces at different primes together to get an exotic example of an $H$-space. This is described quite vividly on p. 79 of Adams's Infinite Loop Spaces.