“I don’t want some spic doctor, I want the other lady!” The patient was 6 years old. He leaned back on the chest of his father, who nodded silently and then agreed: “He would feel more comfortable.” My colleague, a physician-in-training who is from Colombia, stepped out and I took over.

Patients refuse care based on health-care providers’ ethnicity and religion so often that this phenomenon has been dubbed “medicine’s open secret.” A new poll shows that a majority of health-care professionals say they have faced prejudice from patients. In 2013, a nurse in Flint, Mich., sued a pediatric intensive care unit after it granted a request from a father to enter “no African American nurses” on his infant’s care plan. Damon Tweedy, an African American psychiatrist, describes similar experiences in bruising detail throughout his memoir, “Black Man in a White Coat.” And when Esther Choo, an Asian American emergency department physician, tweeted last month that white nationalists refused her care, she set off a Twitter storm of health-care providers responding with similar stories.

Patients have the right to choose their own health-care providers. But two challenging questions emerge when a patient refuses care based on a provider’s religion or ethnicity.