How to tunnel between 2 identical subnets (i.e., both 10.0.0.0/24) ?

I have 2 identical subnets 10.0.0.0/24 at different sites. Both sites have internet access. LAN IP addresses are guaranteed to be unique, like this:
Site A:
10.0.0.1, 10.0.0.3, 10.0.0.5, ...

Site B:
10.0.0.2, 10.0.0.4, 10.0.0.6, ...

I want to slowly migrate all hosts from site A to site B without having to reconfigure them or use different subnets. I do not mind maintaining 2 routers with lists of static IPs which I update as they move, if necessary. All machines must communicate as if they are on the same subnet regardless of site.

For example I have Windows and Linux workstations at both sites. The Windows Domain Controller is at Site A but will eventually move to site B. PCs at both sites must be able to use the DC before and after it moves.

Thanks!