The cohomology of $\mathbb{R}$-motivic $\mathcal{A}(2)$

We compute the cohomology of the quotient algebra $\mathcal{A}(2)$ of the $\mathbb{R}$-motivic dual Steenrod algebra. We do so by running a $\rho$-Bockstein spectral sequence whose input is the cohomology of $\mathbb{C}$-motivic $\mathcal{A}(2)$. The purpose of our computation is that the cohomology of $\mathcal{A}(2)$ is the input to an Adams spectral sequence of a hypothetical $\mathbb{R}$-motivic modular forms spectrum. This Adams spectral sequence computes the homotopy groups of such an $\mathbb{R}$-motivic modular forms spectrum, which in turn can be used to make inferences about the homotopy groups of the $\mathbb{R}$-motivic sphere spectrum and eventually about the classical stable stems.