Is it possible to have aymptomatic asthma?

Yes and no: Asthma in adults is a chronic disease, but many people don't "feel" symptoms until they have a cold, exercise or an allergy or smoke exposure and their lung function drops significantly. Even between attacks the inflammation can be present and lung function below normal. Now there is a breath test called exhaled Nitric Oxide that can help measure lung inflammation even when no symptoms.

Yes: Asthma is not a one size for all condition.Some will have mild cough variant asthma, as the label suggests, their only symptom may be a cough that resists cough meds but responds to asthma meds. At the other end of the spectrum, they may require several meds daily to keep them out of the ICU with bad attacks. These variants may exist within the same family, since it is linked genetically.