Realization and Tensor Product of Vector Spaces Let V & W are complex vector spaces of dimensions m and n respectively, I want to ask how does realization behaves with the tensor product of vector spaces, more precisely $\\$Can we say that "the realization of the tensor product is same as the tensor product of realized vector spaces? " i.e$$(V\otimes_{\mathbb{C}} W)_{\mathbb{R}}\cong V_{\mathbb{R}}\otimes_{\mathbb{R}} W_{\mathbb{R}}$$
Intuitively looks like it might not be true, considering the dimensions don't match (dim LHS=2mn$\neq$4mn=dim RHS).
But since the dimension is exactly half in the LHS, we might expect something more i.e

*

*Whether there is an isomorphism for 2 copies of LHS (as direct sum) to the RHS.

*Whether there is some relation between them in any other way possible.

Kindly help! Any suggestions would be appreciated.
Thanks and regards
 A: The natural relationship here is that there is a quotient map $V_{\mathbb{R}}\otimes_{\mathbb{R}} W_{\mathbb{R}}\to (V\otimes_{\mathbb{C}} W)_{\mathbb{R}}$.  Indeed, note that the map $\mu:V_\mathbb{R}\times W_\mathbb{R}\to (V\otimes_{\mathbb{C}} W)_{\mathbb{R}}$ defined by $\mu(v,w)=v\otimes w$ is $\mathbb{R}$-bilinear, so by the universal property of the tensor product it gives an $\mathbb{R}$-linear map $f:V_{\mathbb{R}}\otimes_{\mathbb{R}} W_{\mathbb{R}}\to (V\otimes_{\mathbb{C}} W)_{\mathbb{R}}$ such that $f(v\otimes w)=v\otimes w$ (note that the "$v\otimes w$" on the two sides here are not the same thing; on the left side it is an element of the tensor product over $\mathbb{R}$ and on the right side it is an element of the tensor product over $\mathbb{C}$).  To see that this $f$ is surjective, note that $V\otimes_\mathbb{C} W$ is generated as a $\mathbb{C}$-vector space by elements of the form $v\otimes w$, and therefore is actually also generated as an $\mathbb{R}$-vector space by such elements since $i(v\otimes w)=(iv)\otimes w$.
Another way to think of this is that to form the tensor product, you take formal linear combinations of elements of the form $v\otimes w$ and then impose relations saying that $\otimes$ is a bilinear operation.  The relations that say $\otimes$ is $\mathbb{R}$-bilinear are a subset of the relations that say $\otimes$ is $\mathbb{C}$-bilinear.  So you could construct the tensor product over $\mathbb{C}$ by first imposing the $\mathbb{R}$-bilinearity relations to get the tensor product over $\mathbb{R}$, and then imposing further relations make $\otimes$ actually $\mathbb{C}$-bilinear and get the tensor product over $\mathbb{C}$.  (Note that over $\mathbb{C}$ you have more formal linear combinations than over $\mathbb{R}$, but this doesn't actually give you any new elements in the tensor product over $\mathbb{C}$ since bilinearity forces $i(v\otimes w)=(iv)\otimes w$.)  So, the tensor product over $\mathbb{C}$ is a quotient of the tensor product over $\mathbb{R}$.
