The analytic Hasse Principle for certain singular intersections of quadrics in $\mathbb{P}^9$

For a pair of quadratic forms with rational coefficients in at least $10$ variables, we prove an asymptotic formula for the number of common zeros under the assumption that the two forms determine a projective variety with exactly two (geometric) singular points defined over an imaginary quadratic field. This extends work of Browning and Munshi with the help of automorphic methods.