WASHINGTON — The Supreme Court on Monday narrowly endorsed reducing California’s cramped prison population by more than 30,000 inmates to fix sometimes deadly problems in medical care, ruling that federal judges retain enormous power to oversee troubled state prisons.

The court said in a 5-4 decision that the reduction is “required by the Constitution” to correct longstanding violations of inmates’ rights. The order mandates a prison population of no more than 110,000 inmates, still far above the system’s designed capacity.

There were more than 143,000 inmates in the state’s 33 adult prisons as of May 11, meaning roughly 33,000 inmates will need to be transferred to other jurisdictions or released.