The city’s landlords have long enlisted the help of screening companies to help weed out prospective tenants with a history of rent troubles, but blacklisted tenants have long been without a way to fight back against inaccuracies on those lists. Tenants might appear on such blacklists, for example, after withholding rent because the landlord neglected to make repairs, even if they won their case in Housing Court. And they might never have known it until being rejected from an apartment building they’d hope to live in. In February, the City Council passed the Tenant Fair Chance Act, which requires landlords, property managers and brokers to disclose which screening company they will use in tenant background checks. As a result, tenants will be able to order their own files from these companies and correct any inaccuracies beginning this summer. How much this will help remains to be seen — a file won’t say that the tenant won the case, just that the tenant appeared in court. “There are a lot of tenants who won’t withhold rent or complain to get repairs done because they’re afraid of getting blacklisted,” said James Fishman, a tenants’ lawyer in a recent class-action suit against a screening company.