  We prove the global $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier
integral operators with amplitudes in $S^0_{1,0} (n,2)$. To achieve this, we
require that the phase function can be written as $(x,\xi,\eta) \mapsto
\phase_1(x,\xi) + \phase_2(x,\eta)$ where each $\phase_j$ belongs to the class
$\Phi^2$ and satisfies the strong non-degeneracy condition. This result extends
that of R. Coifman and Y. Meyer regarding pseudodifferential operators to the
case of Fourier integral operators.
