I`ve added
jboss-logging-jdk.jar
jboss-logging-spi.jar
mod_cluster.jar
to Tomcat lib/ directory.

However, when I start Tomcat, no messages from mod_cluster is displayed.
By debugging I was able to understand what NullLoggerPlugin is used.

Where is a code to initialize logging in Tomcat - org.jboss.modcluster.catalina.ModClusterListener.java:65
however, it is not working because org.jboss.logging.Logger.init() called before it.
By default, system tries to load log4j plugin, which is not available in this environment.
After that, it sets NullLoggerPlugin.

To workaround this, I have added -Dorg.jboss.logging.Logger.pluginClass=org.jboss.logging.jdk.JDK14LoggerPlugin
to java options used when Tomcat is started.

I propose fixing org.jboss.modcluster.catalina.ModClusterListener.java with patch (attaching in next comment).