You are right to be suspicious of a seemingly healthy master/slave replication setup! We were running fine when suddenly we got alerts from check_mk concerning a database that existed on our master that did not exist on our slave... but the master and slave status outputs were good! How unnerving is that? The way to prove integrity of the process is to use checksums to verify the data.
I have seen a lot of chatter on the Internet recommending pt-table-checksum . However, its limitations proved to be too onerous for us to be comfortable with. Most importantly, it requires and even sets statement-based replication (see the pt-table-checksum link). As it says in the mysql 5.6 online documentation, (for row-based replication...) "all changes can be replicated. This is the safest form of replication." There are other disadvantages to statement-based replication that make our developers nervous because some functions cannot be replicated properly; see the doc for a list.
We have already experienced issues with a master and slave using statement-based replication so we're specifically trying to avoid it.
We are going to try mysqlrplsync which specifically mentions that it "works independently of the binary log format (row, statement, or mixed)". It also mentions, however, that gtid-mode must be on and it requires MySQL 5.6.14 and higher... which means, I believe, that the MySQL delivered with RHEL7/CentOS 7 at least is out. You'll need to get the MySQL Community Edition, which is left as an exercise for the reader but you can go here for the packages or here for the repos, including RHEL derivatives and Debian.