14. If the system of equations $\left\{\begin{array}{l}\sin x=m \sin ^{3} y, \\ \cos x=m \cos ^{3} y\end{array}\right.$ has real solutions for $x, y$, then the range of positive real values for $m$ is
$$
\begin{array}{l}
\sin ^{2} x+\cos ^{2} x=m^{2}\left(\sin ^{6} y+\cos ^{6} y\right)=1 \Rightarrow \frac{1}{m^{2}}=\sin ^{6} y+\cos ^{6} y \\
=\sin ^{4} y+\cos ^{4} y-\sin ^{2} y \cos ^{2} y=1-3 \sin ^{2} y \cos ^{2} y=1-\frac{3}{4} \sin ^{2} 2 y \in\left[\frac{1}{4}, 1\right] \Rightarrow m \in[1,2],
\end{array}
$$

Therefore, the range of the positive real number $m$ is $[1,2]$.