  We study the effect of disorder in systems having a non-trivial Euler class.
As these recently proposed multi-gap topological phases come about by braiding
non-Abelian charged band nodes residing between different bands to induce
stable pairs within isolated band subspaces, novel properties may be expected.
Namely, a~modified stability and critical phases under the unbraiding to metals
can arise, when the disorder preserves the underlying $C_2\cal{T}$ or
$\cal{P}\cal{T}$ symmetry on average. Employing elaborate numerical
computations, we verify the robustness of associated topology by evaluating the
changes in the average densities of states and conductivities for different
types of disorders. Upon performing a scaling analysis around the corresponding
quantum critical points we retrieve a universality for the localization length
exponent of $\nu = 1.4 \pm 0.1$ for Euler-protected phases, relating to
two-dimensional percolation models. We generically find that quenched disorder
drives Euler semimetals into critical metallic phases. Finally, we show that
magnetic disorder can also induce topological transitions to quantum anomalous
Hall plaquettes with local Chern numbers determined by the initial value of the
Euler invariant.
