In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s coordinate-free spectra” for various reasons, one of which is that a symmetric monoidal structure is easier to obtain (this was before EKMM or the Hovey-Shipley-Smith paper on symmetric spectra). Thomason writes:

As convincing evidence for this claim, I refer to my talk at the Colloque en l’honneur de Michel Zisman at l’Universite Paris VII in June 1993. There I used this alternate model of stable homotopy to give the first known construction of a smash product which is associative and commutative up to coherent natural isomorphism in the model category. This will be the subject of an article to appear.

Unfortunately, he died later that year. Hence my question:

Did anyone ever work out the construction of the smash product he had in mind?