  Globular cluster (GC) systems of massive galaxies often show a bimodal colour
distribution. This has been interpreted as a metallicity bimodality, created by
a two-stage galaxy formation where the red, metal-rich GCs were formed in the
parent halo and the blue metal-poor GCs were accreted. This interpretation,
however, crucially depends on the assumption that GCs are exclusively old
stellar systems with a linear colour-metallicity relation (CZR). The shape of
the CZR and range of GC ages are currently under debate, because their study
requires high quality spectra to derive reliable stellar population properties.
We determined metallicities with full spectral fitting from a sample of 187 GCs
with high spectral signal-to-noise ratio in 23 galaxies of the Fornax cluster
that were observed as part of the Fornax 3D project. The derived CZR from this
sample is non-linear and can be described by a piecewise linear function with a
break point at ($g - z$) $\sim$ 1.1 mag. The less massive galaxies in our
sample ($M_\ast < 10^{10} M_\odot$) appear to have slightly younger GCs, but
the shape of the CZR is insensitive to the GC ages. Although the least massive
galaxies lack red, metal-rich GCs, a non-linear CZR is found irrespective of
the galaxy mass, even in the most massive galaxies ($M_\ast \geq 10^{11}
M_\odot$). Our CZR predicts narrow unimodal GC metallicity distributions for
low mass and broad unimodal distributions for very massive galaxies, dominated
by a metal-poor and metal-rich peak, respectively, and bimodal distributions
for galaxies with intermediate masses (10$^{10}$ $\leq$ $M_\ast < 10^{11}
M_\odot$) as a consequence of the relative fraction of red and blue GCs. The
diverse metallicity distributions challenge the simple differentiation of GC
populations solely based on their colour.
