Tkinter: How can I constantly run another thread/process alongside the event loop?

Question

I'm trying to have a thread/process that runs alongside my Tkinter event loop and constantly checks to see if a certain amount of time has elapsed since the user was last active so that a screensaver can be displayed.

Unfortunately I keep encountering errors. I tried to use a thread with a while True loop but then the event loop would not work.

from tkinter import *
import screensaver
import time
import threading

def screen_save():
    while True:
        if (time.time() - last_time_active > 30):
           screensaver()

last_time_active = time.time()

def refresh_time():
    global last_time_active
    last_time_active = time.time()

screen = Tk()

button = Button(screen, text="activity", command=refresh_time)

thread = threading.Thread(target=screen_save)
thread.start()

mainloop()

I also tried an approach using .after and recursion but that instantly exceeded the maximum recursion depth despite being instruction to wait 30 seconds each .after call.

from tkinter import *
import screensaver
import time

def screen_save():
    if (time.time() - last_time_active > 30):
       screensaver()
    else:
        screen.after(5000, screen_save)

last_time_active = time.time()

def refresh_time():
    global last_time_active
    last_time_active = time.time()

screen = Tk()

button = Button(screen, text="activity", command=refresh_time)

screen.root(5000, screen_save)

mainloop()

Any advice would be greatly appreciated!

Also, in the minimal version of the code I included, screensaver is a working function I made that displays a working Tkinter screensaver. I just retyped what I essentially did.