Tables for stable homotopy groups of $\mathbb{R}P^{\infty} \wedge \mathbb{R}P^{\infty}$

Bob Bruner and Christian Nassau both have code that can compute the Adams charts efficiently. For example, Bob has a chart for $\mathbb{R}P^2\wedge\mathbb{R}P^2$ at http://www.rrb.wayne.edu/cohom/index.html, and he might be able to do $\mathbb{R}P^\infty\wedge\mathbb{R}P^\infty$ with little effort if you asked him nicely. But I don't know where you would find information about the Adams differentials.

I just wanted to chime in to plug my own Ext calculator (a joint venture with Dexter Chua). You can see an interactive calculation of $\mathbb{R}\mathrm{P}^4\wedge\mathbb{R}\mathrm{P}^4$ here:

Ext chart here.

There's a list of other example modules that you can resolve here: https://hoodmane.github.io/rust_ext/

You can interactively add Adams differentials and it will propagate them using a few basic rules, but you'll mostly need to work them out yourself.