  Topological edge states arise in parity-time ($\mathcal{PT}$)-symmetric
non-unitary quantum dynamics but have so far only been discussed in the
$\mathcal{PT}$-symmetry-unbroken regime. Here we report the experimental
detection of robust topological edge states in one-dimensional photonic quantum
walks with spontaneously broken $\mathcal{PT}$ symmetry, thus establishing the
existence of topological phenomena therein. We theoretically prove and
experimentally confirm that the global Berry phase in non-unitary quantum-walk
dynamics unambiguously defines topological invariants of the system in both the
$\mathcal{PT}$-symmetry-unbroken and broken regimes. As topological edge states
exist in both $\mathcal{PT}$ unbroken and broken regimes, we reveal that
topological phenomena are not driven by $\mathcal{PT}$ symmetry.
