You should turn job control on with set -m in your .bashrc before starting tmux.
bash Bash runs startup files like .bashrc with job control disabled. From shell.cshell.c:
#if defined (JOB_CONTROL) /* Startup files should be run without job control enabled. */ old_job_control = interactive_shell ? set_job_control (0) : 0; #endif
#if defined (JOB_CONTROL)
/* Startup files should be run without job control enabled. */
old_job_control = interactive_shell ? set_job_control (0) : 0;
#endif
Please notice that C-b C-z does not set tmux in the background by itself; absent job control in the upper shell, the SIGTSTP signal tmux sends itself will just cause it to stop; the tmux process will still be in the foreground process group of the terminal (despite the deceptive switching from the alternate screen with the the exit_ca_mode/rmcup/te escape on some terminal emulators like xterm)