If one is running Docker Enterprise with Kubernetes in an on-premises private cloud, is it possible to add clusters in a public cloud like Azure?
On GCP, Anthos is a candidate.
You may have a look on their architecture and see if it fits your needs.
Anthos is advertised in most of the GCP architecture courses and offers integration between GKE and both on-prem(the on-prem cluster must meet some prerequisites or you can use the version provided by Google) and AWS Kubernetes clusters.
Istio is a service mesh and if I understood correctly your requirements, the multiple clusters and multiple networks models could be used.
why don't use rancher for that , you can manage on-premise and GKE AKS EKS or cluster installed in ec2.
it's a great tool for that
This is where Azure Arc can help you to achieve this requirement. However it is in preview stage as of now, i hope soon it will be generally available.
From the DOCS,
You can attach and configure Kubernetes clusters inside or outside of Azure by using Azure Arc-enabled Kubernetes Preview. When a Kubernetes cluster is attached to Azure Arc, it will appear in the Azure portal. It will have an Azure Resource Manager ID and a managed identity. Clusters are attached to standard Azure subscriptions, are located in a resource group, and can receive tags just like any other Azure resource.
cluster API under kubernetes SIG is an open source project which provides declarative APIs and tooling to simplify provisioning, upgrading, and operating multiple Kubernetes clusters.
Cluster API can be extended to support any infrastructure provider (AWS, Azure, vSphere, etc.) or bootstrap provider (kubeadm is default) you need. There is a growing list of supported providers available.
You can use cluster API and build your own custom management plane based on cluster API if vendor provided software is not an option for you.
If you are looking for a vendor provided management plane which can be hosted on prem and can manager life cycle of a on prem kubernetes cluster as well as a cluster on any public cloud provider such as AWS, GCP, Azure then Tanzu Mission Control from VMware is an option. Internally it uses cluster API.
Personally I would not use Anthos or Arc because they seem to be a way to get locked into a specific vendor