Unitary change of variables in real integral

Question

I am interested in the solution of the folliwng multivariate Gaussian integral: \begin{equation} I=\int_{\mathbb R^N} \mathrm d x\; e^{-\frac 12 x^T\Omega x} \end{equation} where $\Omega$ is a complex symmetric matrix. I know from the Autonne-Takagi factorization theorem that a unitary matrix $U$ exists such that \begin{equation} U^T \Omega U =D \in M_{N\times N}(\mathbb R) \end{equation} where $D$ is a diagonal matrix with real nonnegative entries. I would like to make use of this result to perform a change of variables $y= Ux$ s.t. \begin{equation} e^{-\frac 12 (Ux)^T \Omega (Ux)} = e^{-\frac 12 x^T D x} = \prod_{i=1}^N e^{-\frac 12 x_i^2 d_i} \end{equation} where $d_i$ are the (diagonal) entries of $D$. I am concerned about the transformation of the measure of integration, because the transformation $x\mapsto y$ maps a variable in $\mathbb R^N$ onto $\mathbb C^N$. In particular, I don't know if the standard rules for the transformation of the Jacobian apply in this case as initial and final spaces of the transformation are not isomorphic. Therefore, I don't know if I can write \begin{equation} \mathrm d x =\det \left\lvert \frac{\partial x}{\partial y}\right\rvert dy =|\det U|^{-1} dy = dy \end{equation}