TITLE: Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
QUESTION [11 upvotes]: Let $E$ be a ring spectrum and $F$ a connective spectrum. Then we have a convergent Atiyah-Hirzebruch spectral sequence $H_s(F,E_t) \Rightarrow E_{s+t}(F)$. Suppose now that $F$ is also a ring spectrum. Then does the multiplication on the $E^2$ page induce multiplications on all subsequent pages, and do they agree with the multiplication on $E_\ast(F)$?
Bizarrely, the only reference I can find for the multiplicative properties of the Atiyah-Hirzebruch spectral sequence is another MO question, and there is only treated the case of the cohomological AHSS where $F$ is a space, rather than the homological spectral sequence where $F$ is a ring spectrum.

REPLY [5 votes]: Just a brief response to Tyler and John's answers- but not brief enough to fit into a comment.
Here is one way to make the filtered object $\{\tau_{\ge n}E\}$ as structured as you'd like (just spelling out exactly what Tyler indicated).
Consider the $\infty$-category $\mathsf{Fun}(\mathbb{Z}, \mathsf{Sp})$ of filtered spectra, where $\mathbb{Z}$ is regarded as a poset. This has a symmetric monoidal structure coming from Day convolution. We have a colimit functor $\mathsf{Fun}(\mathbb{Z}, \mathsf{Sp}) \to \mathsf{Sp}$ and its right adjoint the 'constant tower' functor. The constant tower functor can be promoted to a symmetric monoidal functor $\mathsf{Sp} \to \mathsf{Fun}(\mathbb{Z}, \mathsf{Sp})$ (this is true in general for Day convolution stuff, but here it's extra true since there's only one colimit preserving symmetric monoidal functor from $\mathsf{Sp}$ to any other stable, presentably symmetric monoidal $\infty$-category anyway...).
Consider the full subcategory $\mathcal{C} \subseteq \mathsf{Fun}(\mathbb{Z}, \mathsf{Sp})$ spanned by the objects $\{E_n\}$ such that $E_n$ is $n$-connective, i.e. $E_n = \tau_{\ge n}E_n$. Staring at the formula for Day convolution, we learn that this subcategory is closed under the symmetric monoidal structure because smashing $n$-connective and $m$-connective thing gets you an $(n+m)$-connective thing (and $k$-connective things are closed under hocolims). 
Now general nonsense says we get a colocalization $\mathsf{Fun}(\mathbb{Z}, \mathsf{Sp}) \to \mathcal{C}$ which is canonically lax symmetric monoidal. (HA.2.2.1.1).
The composite $\mathsf{Sp} \to \mathsf{Fun}(\mathbb{Z}, \mathsf{Sp}) \to \mathcal{C} \to \mathsf{Fun}(\mathbb{Z}, \mathcal{C})$ is then canonically lax symmetric monoidal and is given on objects by $E \mapsto \{\tau_{\ge n}E\}$.
From here you can get whatever you want! For example, this induces a lax symmetric monoidal functor on homotopy categories, which gives you the pairings you needed in Tyler's answer. But it does much more: it also tells you that if you start with $E$, and algebra over any operad $\mathcal{O}$, then the Whitehead tower is canonically a filtered algebra over that operad. 
(This is not the end of the story, I think. I always get confused about this but, if I remember correctly, people often like to ask for some filtered version of the operad to act on this filtered gadget, in order to get the story of power operations in the spectral sequence? I might be confusing this with something else though... again- I never learned that story properly).