  We present the first example of a grand unified theory (GUT) with a modular
symmetry interpreted as a family symmetry. The theory is based on
supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified
on a $T_2/\mathbb{Z}_2$ orbifold. We have shown that, if there is a finite
modular symmetry, then it can only be $A_4$ with an (infinite) discrete choice
of moduli, where we focus on $\tau = \omega=e^{i2\pi/3}$, the unique solution
with $|\tau|=1$. The fields on the branes respect a generalised CP and flavour
symmetry $A_4\ltimes \mathbb{Z}_2$ which is isomorphic to $S_4$ which leads to
an effective $\mu-\tau$ reflection symmetry at low energies, implying maximal
atmospheric mixing and maximal leptonic CP violation. We construct an explicit
model along these lines with two triplet flavons in the bulk, whose vacuum
alignments are determined by orbifold boundary conditions, analogous to those
used for $SU(5)$ breaking with doublet-triplet splitting. There are two
right-handed neutrinos on the branes whose Yukawa couplings are determined by
modular weights. The charged lepton and down-type quarks have diagonal and
hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark
Yukawa matrix.
