Answer
Check whether the external system's own network range overlaps with the cluster's internal pod CIDR — Kubernetes-internal networking is private and won't disturb anything outside the cluster on its own, but if a pod needs to reach an external system whose address range happens to overlap with the cluster's internal pod range, that overlap can cause exactly this kind of routing confusion, and the cluster's pod CIDR may need to be changed to a non-overlapping range.