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