How to get rid of strep throat fast without an antibiotic?

You can't. If you have actually been diagnosed with strep throat (with a test....It can't be accurately diagnosed by looking at it), then it is really important to clear it up with an antibiotic. The bacteria that causes strep throat can cause other problems with your kidneys or heart, and no alternative medicine will prevent those problems from happening.
You don't. Strep throat goes through several phases. The acute phase will eventually fade in most without rx. During the sub-acute phase, the body continues to make antibodies & a fraction of the cases develop long term complications including kidney, heart & brain injury. Antibiotics can stop the sub-acute & complications if started within ~10 days of first fever. Herbal or comfort rx makes no difference.
You shouldn't try. Sore throats mostly go away with or without antibiotics. But if you have a "strep" (specifically a bacterium called s. Pyogenes or group a strep) foregoing antibiotics can occasionally lead to serious complications. I recommend you see your doctor if you haven't. Get tested. If it's a virus, then symptom control is fine. If it's strep, take and finish your antibiotic.