Example 5 Take $m_{1}=3, m_{2}=4, m=m_{1} m_{2}=12$. Then take
$$x_{i}^{(1)}=i, 1 \leqslant i \leqslant 6 ; \quad x_{1}^{(2)}=0, x_{2}^{(2)}=2$$

It is not difficult to calculate that $x_{i}^{(1)}+3 x_{j}^{(2)}(1 \leqslant i \leqslant 6,1 \leqslant j \leqslant 2)$ is a complete residue system modulo 12.