Quadratic forms with common real zeros

Question

Is it possible to find three quadratic forms $f_1,f_2,f_3$ in $K[x,y,z]$ where $K$ is a real number field such that

1. $f_1,f_2$ and $f_3$ have exactly 2 common real zeros
2. $f_1^2+f_2^2+f_3^2$ can be factorized into product of four linear forms
3. $f_1,f_2$ and $f_3$ are $K$ linearly independent in the vector space of quadratic forms over $K$.