GlobalGetEx seems to not work on signature mode. Is there any way that applications can communicate with logicsigs?
Applications cannot read logicsigs state directly.
Logicsigs and logicsig arguments are not signed and may in theory even not be recorded on the blockchain (though they currently are).
Before a transaction is committed, any node on the network can change the arguments of a logicsig as long as the logicsig still approves the transaction.
This is completely different from the arguments of applications that are signed by the sender.
That being said, logicsigs can still communicate with applications via the following:
Logicsigs can validate any other field of any other transaction, so they can check for example that arguments of an application satisfy some logic.
However, logicsigs cannot dynamically generate a value that would be use by an application.