  We use large-scale Monte Carlo simulations to test the Weinrib-Halperin
criterion that predicts new universality classes in the presence of
sufficiently slowly decaying power-law-correlated quenched disorder. While new
universality classes are reasonably well established, the predicted exponents
are controversial. We propose a method of growing such correlated disorder
using the three-dimensional Ising model as benchmark systems both for
generating disorder and studying the resulting phase transition. Critical
equilibrium configurations of a disorder-free system are used to define the
two-value distributed random bonds with a small power-law exponent given by the
pure Ising exponent. Finite-size scaling analysis shows a new universality
class with a single phase transition, but the critical exponents
$\nu_d=1.13(5), \eta_d=0.48(3)$ differ significantly from theoretical
predictions. We find that depending on details of the disorder generation,
disorder-averaged quantities can develop peaks at two temperatures for finite
sizes. Finally, a layer model with the two values of bonds spatially separated
to halves of the system genuinely has multiple phase transitions and
thermodynamic properties can be flexibly tuned by adjusting the model
parameters.
