jmc

Calculate the magnitudes. $$|t+2i\sqrt{3}||6-4i|=\sqrt{t^2+12}\cdot \sqrt{36+16}=\sqrt{t^2+12}\cdot \sqrt{52}=\sqrt{t^2+12}\cdot 2\sqrt{13}$$Set this equal to $26$. $$\sqrt{t^2+12}\cdot 2\sqrt{13}=26$$Solve for $t$. $$\sqrt{t^2+12}\cdot \sqrt{13}=13$$$$\sqrt{t^2+12}=\sqrt{13}$$We need the positive value, so $t=\boxed{1}$.