Hot-Hot Datacenters with DNS LTM APM SSO

You cannot have APM active-active as stated in K13983: BIG-IP APM HA deployments

...you must deploy HA configurations for APM traffic groups in an active-standby-standby (N + M) topology.

You can use F5 DNS and set persistence to try and ensure once a user is sent to a DC they will stick to it, however this has its problems if using public DNS servers and you will likely find users will flip due to ISP DNS configuration which is out of your control.

So your only real option it you want true active-active data centres, and not rely on F5 DNS persistence, is to look into a data centre affinity solution.

Best option is for you to set a cookie with a value related to each DC and have LTM query this cookie value in an iRule or Policy and if it doesn't match it local value to forward to the other DC.

The difficult bit is making sure you only forward traffic when required and not if users have been forwarded due to a failure in the one DC. So need a method of checking availability of the remote DC Virtual Server/s you will be forwarding too.