  Hundreds of millions of supermassive black hole binaries are expected to
contribute to the gravitational-wave signal in the nanohertz frequency band.
Their signal is often approximated either as an isotropic Gaussian stochastic
background with a power-law spectrum, or as an individual source corresponding
to the brightest binary. In reality, the signal is best described as a
combination of a stochastic background and a few of the brightest binaries
modeled individually. We present a method that uses this approach to
efficiently create realistic pulsar timing array datasets using synthetic
catalogs of binaries based on the Illustris cosmological hydrodynamic
simulation. We explore three different properties of such realistic backgrounds
which could help distinguish them from those formed in the early universe: i)
their characteristic strain spectrum; ii) their statistical isotropy; and iii)
the variance of their spatial correlations. We also investigate how the
presence of confusion noise from a stochastic background affects detection
prospects of individual binaries. We calculate signal-to-noise ratios of the
brightest binaries in different realizations for a simulated pulsar timing
array based on the NANOGrav 12.5-year dataset extended to a time span of 15
years. We find that $\sim$6% of the realizations produce systems with
signal-to-noise ratios larger than 5, suggesting that individual systems might
soon be detected (the fraction increases to $\sim$41% at 20 years). These can
be taken as a pessimistic prediction for the upcoming NANOGrav 15-year dataset,
since it does not include the effect of potentially improved timing solutions
and newly added pulsars.
