Setup
Suppose
$$ dZ_t^{(1)}=X_t^{(1)}dt+Y_t^{(1)}dB_t, \qquad dZ_t^{(2)}=X_t^{(2)}dt+Y_t^{(2)}dB_t. $$Formula
Then
$$ d(Z_t^{(1)}Z_t^{(2)})
Z_t^{(1)},dZ_t^{(2)} + Z_t^{(2)},dZ_t^{(1)} + d\langle Z^{(1)},Z^{(2)}\rangle_t. $$
Cross variation
The cross variation satisfies
$$ d\langle Z^{(1)},Z^{(2)}\rangle_t
Y_t^{(1)}Y_t^{(2)},dt. $$
Check
If $Z^{(1)}=Z^{(2)}=Z$, then the rule reduces to
$$ d(Z_t^2)=2Z_t\,dZ_t+d\langle Z\rangle_t, $$which matches Ito formula for $f(z)=z^2$.