Let $U \subset \mathbb{C}^n$ an open domain and $F \colon U \to \mathbb{C}^n$ a holomorphic map. Consider the autonomous differential equation
$$ \frac{dz}{dt} = F(z), \qquad (t,z) \in \mathbb{C} \times U. $$
As far as I know, it is a standard result that for any $(t_0,z_0) \in \mathbb{C} \times U$ there exist a polydisk
$$ D_\varepsilon = \{ |t-t_0| < \varepsilon, |z_j-z_{0,j}| < \varepsilon, j=1,\dots,n\} \subset \mathbb{C} \times U $$
for sufficiently small $\varepsilon > 0$, such that the solution exists and is unique in $D_\varepsilon$. Moreover, the solution depends holomorphically on initial data, that is to say the flow $\Phi\colon (t,z) \to \Phi_t(z)$ generated by the vector field $F$ is holomorphic in both $t$ and $z$.
My question is: does this result still hold when $t$ is a real parameter? In particular I'm interested on the holomorphic dependence on initial data, i.e. is $\Phi_t\colon \mathbb{C}^n \to \mathbb{C}^n$ a holomorphic map for $t \in I \subset \mathbb{R}$?