If your RTC have an SQW output and if you can set this output to 1Hz, you could add atach an interrupt to this pin and have your arduino increase a variable, and each time the variable reach 10 (at 1Hz that would mean 10 seconds elapsed), do your stuff, and don't forget to reset that variable to 0.
I know the DS1307, 3231, 3232, 3234 are capable of doing this, probably many others but check with the datasheet of your RTC.