I need to clean my bathroom tiles, but I don't want to use any of those special cleaning liquids. Is there a way to clean bathroom tiles using common household products?
To clean without chemicals you'll want to explore options that do not contain matter.
One possibility is to clean using sound. There are different options available, but they all work on the general premise of dislodging dust, oil, and other particles using rapid sound induced pressure fluctuations. This technique has been used for decades over a wide range of uses. Ultrasonic cleaning uses sound waves that are outside of the human auditory range (your neighbors will be happy!) and is used for cleaning things like jewelry and surgical instruments. Acoustic cleaning uses sound waves that humans can hear and are effective at cleaning larger items like grain elevators.
Electromagnetic radiation provides many options, including radio waves, visible light, ultraviolet light, x-rays, and gamma rays. It's not uncommon to find cleaning products that use ultraviolet light to kill or inactivate bacteria, viruses, and other microorganisms, but I'd recommend for your own safety that you steer clear of x-rays and gamma rays (despite their increased effectiveness).
Avoid any kind of soap, as all of them (even the "natural" ones) contain dozens of chemicals. Vinegar should also be avoided, as it is a combination of water and the chemical acetic acid. You'll also want to avoid water, too, since it's made of the chemical dihydrogen monoxide. Just remember, if it's made out of matter it's made out of chemicals.