For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U_i$, we can build a "descent spectral sequence" with signature $$E^1_{p, q} = \pi_{p+q} \mathcal{F}\left(\coprod_{|I| = q} U_I \right) \Rightarrow \pi_{p+q} \mathcal{F}(X).$$ This comes about by building the simplicial object associated to the cover, applying the sheaf to get a cosimplicial object, and then instantiating a filtration-type spectral sequence given by looking at varying truncations of totalizations --- the standard spectral sequence for a cosimplicial object. The $E^2$-page of the spectral sequence can be identified with Cech cohomology, and so the spectral sequence is meant to provide an intermediary between that homological object and the homotopy-sensitive information in the sheaf of spectra. This construction is natural enough that...

- ...a refinement of the covering induces a map of spectral sequences. Just like limiting Cech cohomologies over cover refinements gives sheaf cohomology, limiting Cech descent spectral sequences gives a descent spectral sequence with $E^2$-page described by sheaf cohomology.
- ...a map of sheaves induces a map of spectral sequences.

This construction doesn't really use the fact that $\mathcal{F}$ takes values in rings. I feel that this must appear in the spectral sequence, that we should expect some kind of multiplicativity --- maybe one that mixes the products of the ring spectra and the Cech product.

So, my question is:Is there a multiplicative structure in any of these spectral sequences? What is its signature? How might I compute with it? Better yet, are there examples of other people computing with it that I can read about?

For what it's worth, I'm interested most in starting with a sheaf of ring spectra $\mathcal{F}$ and computing $\mathcal{F}(X)^*(Y)$ for my favorite space $Y$ by augmenting $\mathcal{F}$ to $F(\Sigma^\infty_+ Y, \mathcal{F})(U) := F(\Sigma^\infty_+ Y, \mathcal{F}(U))$ and working with that. This comes with a map $F(\Sigma^\infty_+ (Y \times Y), \mathcal{F}) \to F(\Sigma^\infty_+ Y, \mathcal{F})$, which is in the vein of the usual construction of the cup product.

(There are a lot of details I'm eliding past, like conditional convergence, $\lim^1$ problems, when my augmented guy is actually a homotopy sheaf, when the analogy with sheaf cohomology can be made, so on and so forth. I don't think I've said anything not true in the nicest of settings, which is where I'd like to start learning, at least. Sorry if this is off-putting.)