  Cosmological $N$-body simulations of the dark matter component of the
universe typically use initial conditions with a fixed power spectrum and
random phases of the density field, leading to structure consistent with the
local distribution of galaxies only in a statistical sense. It is, however,
possible to infer the initial phases which lead to the configuration of
galaxies and clusters that we see around us. We analyse the CSiBORG suite of
101 simulations, formed by constraining the density field within 155 Mpc$/h$
with dark matter particle mass $4.38\times10^9 M_\odot$, to quantify the degree
to which constraints imposed on 2.65 Mpc$/h$ scales reduce variance in the halo
mass function and halo-halo cross-correlation function on a range of scales.
This is achieved by contrasting CSiBORG with a subset of the unconstrained
Quijote simulations and expectations for the $\Lambda$CDM average. Using the
FOF, PHEW and HOP halofinders, we show that the CSiBORG suite beats cosmic
variance at large mass scales ($\gtrsim 10^{14}M_{\odot}/h$), which are most
strongly constrained by the initial conditions, and exhibits a significant
halo-halo cross-correlation out to $\sim30$ Mpc$/h$. Moreover, the effect of
the constraints percolates down to lower mass objects and to scales below those
on which they are imposed. Finally, we develop an algorithm to "twin" halos
between realisations and show that approximately 50% of halos with mass greater
than $10^{15}M_{\odot}/h$ can be identified in all realisations of the CSiBORG
suite. We make the CSiBORG halo catalogues publicly available for future
applications requiring knowledge of the local halo field.
