  Following Ribet's seminal 1976 paper there have been many results employing
congruences between stable cuspforms and lifted forms to construct non-split
extensions of Galois representations. We show how this strategy can be extended
to construct elements in the Bloch-Kato Selmer groups of +/--Asai (or tensor
induction) representations associated to Bianchi modular forms. We prove, in
particular, how the Galois representation associated to a suitable low weight
Siegel modular form produces elements in the Selmer group for exactly the Asai
representation (+ or -) that is critical in the sense of Deligne.
  We further outline a strategy using an orthogonal-symplectic theta
correspondence to prove the existence of such a Siegel modular form and explain
why we expect this to be governed by the divisibility of the near-central
critical value of the Asai L-function, in accordance with the Bloch-Kato
conjecture.
