  Spectral rigidity in Hermitian quantum chaotic systems signals the presence
of dynamical universal features at timescales that can be much shorter than the
Heisenberg time. We study the analog of this timescale in many-body
non-Hermitian quantum chaos by a detailed analysis of long-range spectral
correlators. For that purpose, we investigate the number variance and the
spectral form factor of a non-Hermitian $q$-body Sachdev-Ye-Kitaev (nHSYK)
model, which describes $N$ fermions in zero spatial dimensions. After an
analytical and numerical analysis of these spectral observables for
non-Hermitian random matrices, and a careful unfolding, we find good agreement
with the nHSYK model for $q > 2$ starting at a timescale that decreases sharply
with $q$. The source of deviation from universality, identified analytically,
is ensemble fluctuations not related to the quantum dynamics. For fixed $q$ and
large enough $N$, these fluctuations become dominant up until after the
Heisenberg time, so that the spectral form factor is no longer useful for the
study of quantum chaos. In all cases, our results point to a weakened or
vanishing spectral rigidity that effectively delays the observation of full
quantum ergodicity. We also show that the number variance displays
nonstationary spectral correlations for both the nHSYK model and random
matrices. This nonstationarity, also not related to the quantum dynamics,
points to intrinsic limitations of these observables to describe the quantum
chaotic motion. On the other hand, we introduce the local spectral form factor,
which is shown to be stationary and not affected by collective fluctuations,
and propose it as an effective diagnostic of non-Hermitian quantum chaos. For
$q = 2$, we find saturation to Poisson statistics at a timescale of $\log D$,
compared to a scale of $\sqrt D$ for $ q>2$, with $D $ the total number of
states.
