  We investigate the connection between the star formation rate (SFR) of
galaxies and their central black hole accretion rate (BHAR) using the EAGLE
cosmological hydrodynamical simulation. We find, in striking concurrence with
recent observational studies, that the <SFR>--BHAR relation for an AGN selected
sample produces a relatively flat trend, whilst the <BHAR>--SFR relation for a
SFR selected sample yields an approximately linear trend. These trends remain
consistent with their instantaneous equivalents even when both SFR and BHAR are
time-averaged over a period of 100~Myr. There is no universal relationship
between the two growth rates. Instead, SFR and BHAR evolve through distinct
paths that depend strongly on the mass of the host dark matter halo. The
galaxies hosted by haloes of mass M200 $\lesssim 10^{11.5}$Msol grow steadily,
yet black holes (BHs) in these systems hardly grow, yielding a lack of
correlation between SFR and BHAR. As haloes grow through the mass range
$10^{11.5} \lesssim$ M200 $\lesssim 10^{12.5 }$Msol BHs undergo a rapid phase
of non-linear growth. These systems yield a highly non-linear correlation
between the SFR and BHAR, which are non-causally connected via the mass of the
host halo. In massive haloes (M200 $\gtrsim 10^{12.5}$Msol) both SFR and BHAR
decline on average with a roughly constant scaling of SFR/BHAR $\sim 10^{3}$.
Given the complexity of the full SFR--BHAR plane built from multiple
behaviours, and from the large dynamic range of BHARs, we find the primary
driver of the different observed trends in the <SFR>--BHAR and <BHAR>--SFR
relationships are due to sampling considerably different regions of this plane.
