Not really an answer, but instead a remark. Over the category of smooth real manifolds the symmetric power of smooth curves is not smooth in general. The second symmetric power is a smooth surface with boundary and starting from the third symmetric power what we get are varieties with corners at the boundary. 

Symmetric powers of smooth surfaces are still smooth as they are locally diffeomorphic to complex curves.  

If nothing else these examples show that smoothness might mean different things <strike>for algebraic geometers and differential geometers</strike> in algebraic geometry over $\mathbb R$ and in differential geometry.