Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper

I think that the best thing in this direction is the paper "Permutative categories, multicategories and algebraic K-theory" by Elmendorff and Mandell.

I have only skimmed this so I may well not be understanding it correctly.

Anyway, we want to consider symmetric monoidal categories, whose monoidal structure should be thought of as "addition", so I will call them "plus-monoidal categories". We want to define a kind of tensor product of plus-monoidal categories. Elmendorff and Mandell say that we should first generalise and consider plus-multicategories. We can then define multilinear functors between plus-multicategories (as at https://ncatlab.org/nlab/show/multicategory), making the class of plus-multicategories into a tensor-multicategory. One can then show that there are universal multilinear functors in a suitable sense, so we can form tensor products of plus-multicategories as desired.

As for connective spectra, a good smash product was first worked out by Manos Lydakis in the context of $\Gamma$-spaces. Thomason came to Bielefeld in the mid 1990s and spoke with Waldhausen's group about this matter. Lydakis carried it through.