In this situation, there are 2 main actions a physician may take (I am biased towards one):
1) Prescribe antibiotics until throat cultures are normal
2) Do nothing
Personally, if a patient is without throat symptoms and has no history of rheumatic fever or kidney damage, I would not have even bothered obtaining a strep test. What for???
Also, a person can be a carrier for strep without suffering any health problems. As such, even if the strep test is positive, but if the patient has no symptoms, I do not recommend treatment. (Which again begs the question of why bother getting a strep test if no treatment will be recommended regardless of the test result.)
I would go so far even to say follow-up cultures are NOT necessary after antibiotic treatment for strep throat if a patient does not have any more symptoms and exam is normal.
Which is why I find it surprising when children and adolescent patients receive multiple courses of antibiotics when they feel perfectly fine, but have received treatment just because a strep test came back positive.
Of course... that's just my opinion as I do acknowledge that there's another school of thought which supports antibiotic treatment of all strep positive cultures with follow-up cultures to ensure eradication.
However, according to the 2012 clinical practice guidelines published by the Infectious Diseases Society of America (statement 12),
"We recommend that GAS carriers do not ordinarily justify efforts to identify them nor do they generally require antimicrobial therapy because GAS carriers are unlikely to spread GAS pharyngitis to their close contacts and are at little or no risk for developing suppurative or nonsuppurative complications (eg, acute rheumatic fever; strong, moderate)."Depending on the doctor you see, you will get different opinions. But it is good for patients to be aware of the varying views on this topic given how common sore throats are.
Clinical Practice Guideline for the Diagnosis and Management of Group A Streptococcal Pharyngitis: 2012 Update by the Infectious Diseases Society of America. Clin Infect Dis. (2012) 55 (10): e86-e102. doi: 10.1093/cid/cis629