  Let $f$ and $g$, of weights $k'>k\geq 2$, be normalised newforms for
$\Gamma_0(N)$, for square-free $N>1$, such that, for each Atkin-Lehner
involution, the eigenvalues of $f$ and $g$ are equal. Let $\lambda\mid\ell$ be
a large prime divisor of the algebraic part of the near-central critical value
$L(f\otimes g,\frac{k+k'-2}{2})$. Under certain hypotheses, we prove that
$\lambda$ is the modulus of a congruence between the Hecke eigenvalues of a
genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$
(vector-valued in general), and a non-endoscopic genus-two cusp form. In
pursuit of this we also give a precise pullback formula for a genus-four
Eisenstein series, and a general formula for the Petersson norm of a Yoshida
lift.
  Given such a congruence, using the 4-dimensional $\lambda$-adic Galois
representation attached to a genus-two cusp form, we produce, in an appropriate
Selmer group, an element of order $\lambda$, as required by the Bloch-Kato
conjecture on values of $L$-functions.
