Can a third element manage the public VIP when nodes are on two different subnets?

An ADC solves the problem of nodes in two different data centers with no common subnets.

The ADC is a third element that manages the public VIP, which is the front network point to the two (or more) individual nodes in the cluster.

See the two diagrams in HA Deployment Configuration Selection for a view of a Layer 2 Adjacent failover, and failover with a third-party ADC. Using Layer 2 Adjacency is simpler; it has fewer elements, making it more reliable, robust, easier to maintain, and it also provides a faster failover convergence time.

An ADC is a more complex solution with more elements to implement, configure, troubleshoot, and maintain. In addition, this complex system requires more failover convergence time. Furthermore, there is a significant investment of time, money, and effort to implement a complex ADC solution, as this is a major endeavor.

Failover convergence time is counted from the moment that something goes wrong with the master server, to the time that the new master server is in a steady state and able to receive and service all the connections that were broken from the failure. This convergence time is always longer with an ADC, all factors being the same, than the simpler Layer 2 Adjacency system failover.