  The Dissipative Spectral Form Factor (DSFF), recently introduced in
[arXiv:2103.05001] for the Ginibre ensemble, is a key tool to study universal
properties of dissipative quantum systems. In this work we compute the DSFF for
a large class of random matrices with real or complex entries up to an
intermediate time scale, confirming the predictions from [arXiv:2103.05001].
The analytic formula for the DSFF in the real case was previously unknown.
Furthermore, we show that for short times the connected component of the DSFF
exhibits a non-universal correction depending on the fourth cumulant of the
entries. These results are based on the central limit theorem for linear
eigenvalue statistics of non-Hermitian random matrices [arXiv:2002.02438,
arXiv:1912.04100].
