Cream of tartar is a buffered acid, be carefull what effect your added level has to pH. Keep well above 4.6. Adding corn syrup is easiest. How much you need depends on 3 factors as they control crystallization.
- What is your final moisture level. One percent moisture will increase or decrease the syrup phase by 3%.
- What is the ratio of sucrose to other sugars. From the milk (lactose) and any added corn syrup solids.
- What is the final viscosity. More viscous requires less other sugars to doctor.
Also the colder it's stored the less sugar can stay in solution.
A traditional dulce de leche is just sugar and milk. The cooking time is long, which continues to invert some of the sugar. Same thing the cream of tartar does but much slower. High moisture versions (pudding or flan like) are kept refrigerated as their Aw is too high for good shelf life. Versions with lower moisture can be grained unless they have additional sugars (invert, corn syrup) added to increase the total solids the moisture can hold at ambient temperatures.