  We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class which have
polynomial Euler product and satisfy Selberg's orthonormality condition. We
show that on every vertical line $s=\sigma+it$ with $\sigma\in(1/2,1)$, these
$L$-functions simultaneously take large values of size $\exp\left(c\frac{(\log
t)^{1-\sigma}}{\log\log t}\right)$ inside a small neighborhood.
