I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R}^n,U(\mathbb{C}^n))$ be the subspace consisting of those maps which are adjoint to equivalences $\mathbb{R}^n \otimes \mathbb{C} \to \mathbb{C}^n$. Then $C$ is a connected component of the mapping space, and since the connected group $GL(U(\mathbb{C}^n))$ acts on this mapping space by post-composition it must preserve this component. On the other hand, if one restricts this action to $GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n))$ then $C$ becomes a principal homogeneous space. This implies that for every $n$ the subspace $GL_n(\mathbb{C}) \cong GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n)) \cong GL_{2n}(\mathbb{R})$ is a retract up to homotopy. At $n=1$ this happens to be ok, as $GL_1(\mathbb{C}) \simeq S^1$ is indeed a retract of $GL_2(\mathbb{R}) \simeq O(2) \simeq S^1 \coprod S^1$, but I would bet you would be able to find an $n$ where this inclusion is not a homotopy retract.

I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R}^n,U(\mathbb{C}^n))$ be the subspace consisting of those maps which are adjoint to equivalences $\mathbb{R}^n \otimes \mathbb{C} \to \mathbb{C}^n$. Then $C$ is a connected component of the mapping space, and since the connected group $GL(U(\mathbb{C}^n))$ acts on this mapping space by post-composition it must preserve this component. On the other hand, if one restricts this action to $GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n))$ then $C$ becomes a principal homogeneous space. This implies that for every $n$ the subspace $GL_n(\mathbb{C}) \cong GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n)) \cong GL_{2n}(\mathbb{R})$ is a retract up to homotopy. At $n=1$ this happens to be ok, as $GL_1(\mathbb{C}) \simeq S^1$ is indeed a retract of $GL_2(\mathbb{R}) \simeq O(2) \simeq S^1 \coprod S^1$, but I would bet you would be able to find an $n$ where this inclusion is not a homotopy retract.

-- Edit --

As pointed out in the comments below:

1. Already when $n=2$ the map $GL(\mathbb{C}^n) \to GL(U(\mathbb{C}^n))$ is not a homotopy retract.

2. $GL(U(\mathbb{C}^n))$ is not connected, but one can replace it with the connected component of the identify $GL^0(U(\mathbb{C}^n)) \subseteq GL(U(\mathbb{C}^n))$, replace $C$ with the corresponding $C^0 \subseteq C$, and continue the argument as before (since $GL(\mathbb{C}^n)$ is connected its image in $GL(U(\mathbb{C}^n))$ lies in $GL^0(U(\mathbb{C}^n))$, and the inclusion $GL(\mathbb{C}^n) \hookrightarrow GL^0(U(\mathbb{C}^n))$ is again not a retract already for $n=2$).

Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R}^n,U(\mathbb{C}^n))$ be the subspace consisting of those maps which are adjoint to equivalences $\mathbb{R}^n \otimes \mathbb{C} \to \mathbb{C}^n$. Then $C$ is a connected component of the mapping space, and since the connected group $GL(U(\mathbb{C}^n))$ acts on this mapping space by post-composition it must preserve this component. On the other hand, if one restricts this action to $GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n))$ then $C$ becomes a principal homogeneous space. This implies that for every $n$ the subspace $GL_n(\mathbb{C}) \cong GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n)) \cong GL_{2n}(\mathbb{R})$ is a retract up to homotopy. At $n=1$ this happens to be ok, as $GL_1(\mathbb{C}) \simeq S^1$ is indeed a retract of $GL_2(\mathbb{R}) \simeq O(2) \simeq S^1 \coprod S^1$, but I would bet you would be able to find an $n$ where this inclusion is not a homotopy retract.

I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R}^n,U(\mathbb{C}^n))$ be the subspace consisting of those maps which are adjoint to equivalences $\mathbb{R}^n \otimes \mathbb{C} \to \mathbb{C}^n$. Then $C$ is a connected component of the mapping space, and since the connected group $GL(U(\mathbb{C}^n))$ acts on this mapping space by post-composition it must preserve this component. On the other hand, if one restricts this action to $GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n))$ then $C$ becomes a principal homogeneous space. This implies that for every $n$ the subspace $GL_n(\mathbb{C}) \cong GL(\mathbb{C}^n) \hookrightarrow GL(U(\mathbb{C}^n)) \cong GL_{2n}(\mathbb{R})$ is a retract up to homotopy. At $n=1$ this happens to be ok, as $GL_1(\mathbb{C}) \simeq S^1$ is indeed a retract of $GL_2(\mathbb{R}) \simeq O(2) \simeq S^1 \coprod S^1$, but I would bet you would be able to find an $n$ where this inclusion is not a homotopy retract.