6. Given that $F_{1}, F_{2}$ are the common foci of an ellipse and a hyperbola, $P$ is one of their common points, and $\angle F_{1} P F_{2}=60^{\circ}$, then the minimum value of the product of the eccentricities of the ellipse and the hyperbola is
A. $\frac{\sqrt{3}}{3}$
B. $\frac{\sqrt{3}}{2}$
C. 1
D. $\sqrt{3}$
Let the length of the major axis of the ellipse be $2 a_{1}$, the length of the real axis of the hyperbola be $2 a_{2}$, and their focal distance be $2 c$. Let $\left|P F_{1}\right|=x,\left|P F_{2}\right|=y$, then 
\[
\left\{\begin{array}{l}
x+y=2 a_{1}, \\
x-y=2 a_{2}, \\
x^{2}+y^{2}-x y=4 c^{2}
\end{array} \Rightarrow a_{1}^{2}+3 a_{2}^{2}=4 c^{2} \Rightarrow \frac{1}{e_{1}^{2}}+\frac{3}{e_{2}^{2}}=4\right.
\]
Thus, we have $4=\frac{1}{e_{1}^{2}}+\frac{3}{e_{2}^{2}} \geqslant \frac{2 \sqrt{3}}{e_{1} e_{2}} \Rightarrow e_{1} e_{2} \geqslant \frac{\sqrt{3}}{2}$, so the answer is $B$.