So say you have two mySQL servers called A and B. A and B are in a circular replication ring meaning
A replicates from B
B replicates from A
In addition, log-slave-updates is on, so if one server of either A or B goes down you can recover all the data from either side.
Therefore, here are some steps.
Take a snapshot of a server in the ring, with your favorite backup tool (mine is ibbackup)
When either server A or B dies, restore from said backup. Here are the steps. Let us assume server A goes down, and backups are done from server B.
restore the backup of server B to server A
On server-A turn off log-slave-updates (you will see why on the next line)
Add replicate-same-server-id on server A to my.cnf
CHANGE MASTER TO to the position and file of the binary log that is reported from your tool on server-A.
IF the binary log did not get corrupted on server-B your cool, else flip the binary log to the next binary log from server-A on server-B.
Let server-A catch up
Add log-slave-updates on server-A
remove replicate-same-server-id on server A