  We study the geometrical features of the order parameter's fluctuations near
the critical point of mixed-order phase transitions in randomly interdependent
spatial networks. In contrast to continuous transitions, where the structure of
the order parameter at criticality is fractal, in mixed-order transitions the
structure of the order parameter is known to be compact. Remarkably, we find
that although being compact, the fluctuations of the order parameter close to
mixed-order transitions are fractal up to a well-defined correlation length
$\xi'$, which diverges when approaching the critical threshold. We characterize
the self-similar nature of these critical fluctuations through their fractal
dimension, $d_f'=3d/4$, and correlation length exponent, $\nu'=2/d$, where $d$
is the dimension of the system. By means of percolation and magnetization, we
demonstrate that $d_f'$ and $\nu'$ are independent on the symmetry of the
underlying process for any $d$ of the underlying networks.
