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In this paper, we show that the finite subalgebra$\mathcal{A}^{\mathbb{R}}(1)$, generated by $\mathrm{Sq}^1$ and$\mathrm{Sq}^2$, of the $\mathbb{R}$-motivic Steenrod algebra$\mathcal{A}^{\mathbb{R}}$ can be given $128$ different$\mathcal{A}^{\mathbb{R}}$-module structures. We also show that all of these$\mathcal{A}^{\mathbb{R}}$-modules can be realized as the cohomology of a$2$-local finite $\mathbb{R}$-motivic spectrum. The realization results areobtained using an $\mathbb{R}$ -motivic analogue of the Toda realizationtheorem. We notice that each realization of $\mathcal{A}^{\mathbb{R}}(1)$ canbe expressed as a cofiber of an $\mathbb{R}$-motivic $v_1$-self-map. The$\mathrm{C}_2$-equivariant analogue of the above results then follows becauseof the Betti realization functor. We identify a relationship between the$\mathrm{RO}(\mathrm{C}_2)$-graded Steenrod operations on a$\mathrm{C}_2$-equivariant space and the classical Steenrod operations on bothits underlying space and its fixed-points. This technique is then used toidentify the geometric fixed-point spectra of the $\mathrm{C}_2$-equivariantrealizations of $\mathcal{A}^{\mathrm{C}_2}(1)$. We find another application ofthe $\mathbb{R}$-motivic Toda realization theorem: we produce an$\mathbb{R}$-motivic, and consequently a $\mathrm{C}_2$-equivariant, analogueof the Bhattacharya-Egger spectrum $\mathcal{Z}$, which could be of independentinterest.

On realizations of the subalgebra $A^R(1)$ of the $R$-motivic Steenrod Algebra