I do not know with certainty, but I think the isomerization would proceed in a route similar to below:
NH2C(=S)NH2 --> NH2C(=S)NH(-) H(+) --> NH2C[--S(-)]=NH H(+) --> (+)NH3C[--S(-)]=NH
--> NH3 S=C=NH --> NH4(+) (-)N=C=S
May you find my post a satisfactory answer