  We obtain an explicit classification of all $2\times 2$ real hypergeometric
Bochner pairs, ie. pairs $(W(x),\mathfrak{D})$ consisting of a $2\times 2$ real
hypergeometric differential operator $\mathfrak{D}$ and a $2\times 2$ weight
matrix satisfying the property that $\mathfrak{D}$ is symmetric with respect to
the matrix-valued inner product defined by W(x). Furthermore, we obtain a
classifying space of hypergeometric Bochner pairs by describing a bijective
correspondence between the collection of pairs and an open subset of a real
algebraic set whose smooth paths correspond to isospectral deformations of the
weight W(x) preserving a bispectral property. We also relate the hypergeometric
Bochner pairs to classical Bochner pairs via noncommutative bispectral Darboux
transformations.
