  Magnetotransport measurements on the centrosymetric square-net
Eu(Ga$_{1-x}$Al$_x$)$_4$ compounds reveal evidence for both reciprocal- and
real-space topology. For compositions $0.50 \leq x \leq 0.90$, several
intermediate field phases are found by magnetization measurements when $H
\parallel c$, where a maximum in the topological Hall effect (THE) is observed,
pointing to the existence of topological (real-space topology) or non-coplanar
spin textures. For $0.25 \leq x \leq 0.39$, magnetization measurements reveal
an intermediate field state, but no transition is visible in the Hall
measurements. For $x = 0.15$, only one magnetic transition occurs below the
N\'eel temperature $T_N$, and no intermediate field spin reorientations are
observed. The Hall effect varies smoothly before the spin-polarized (SP) state.
However, in the SP state, Hall measurements reveal a large anomalous Hall
effect (AHE) for all compositions, a consequence of reciprocal-space topology.
Density functional theory calculations in the paramagnetic state indeed reveal
a Dirac point that lies very near the Fermi energy, which is expected to split
into Weyl nodes in the SP state, thereby explaining the observed AHE. These
results suggest the Eu(Ga$_{1-x}$Al$_x$)$_4$ family is a rare material platform
where real- and reciprocal-space topology exist in a single material platform.
