If a diameter of a hyperbola meets the hyperbola in real points then 

Equation of the hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ so equation of the conjugate hyperbola is  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$

Equation of a diameter of the hyperbola is $y=\left(b^2/a^{2}m\right)x$ which meets the hyperbola at points given by

$\frac{x^2}{a^2}-\frac{b^4{x}^2}{b^2{a}^4{m}^2}=1\Rightarrow x^2=\frac{a^2{m}^2}{m^2-b^2}$

These points are real if $m^2>b^2$ and for this value of m, the diameter will meet the conjugate hyperbola at  point for which $x^2=\frac{-a^2{m}^2}{m^2-b^2}$ and these points are  imaginary.

Note: Conjugate diameter will meet the conjugate hyperbola in real points and the hyperbola in imaginary points