I think you need both the Multi-AZ and the Read Replica features of AWS RDS.
Multi-AZ just creates a non-accessible secondary DB in another availability zone and in case the primary fails, AWS would switch over to the secondary DB. So you have failover.
In the case you want to increase the performance, and your application can work in read-only mode in Singapore (for example), the Read Replica would be perfect. If writes are also required, you would need to route them to the primary read-write database.
AWS supports a combination of the two approaches.