  We introduce the notion of the algebraic overshear density property which
implies both the algebraic notion of flexibility and the holomorphic notion of
the density property. We investigate basic consequences of this stronger
property, and propose further research directions in this borderland between
affine algebraic geometry and elliptic holomorphic geometry. As an application,
we show that any smoothly bordered Riemann surface with finitely many boundary
components that is embedded in a complex affine surface with the algebraic
overshear density property admits a proper holomorphic embedding.
