What do you wear on a visit to one of the world’s most notorious prisons?

If you’re Kristi Noem, the homeland security secretary who visited El Salvador’s massive Terrorism Confinement Center on Wednesday, the answer was a white long-sleeve top, gray slacks and a baseball cap emblazoned with the Immigration and Customs Enforcement logo.

Oh, and a gold Rolex Cosmograph Daytona that sells for about $50,000.

Ms. Noem traveled to the prison, known as Cecot, where the Trump administration this month sent hundreds of Venezuelan deportees. Earlier this week, the U.S. Court of Appeals for the District of Columbia Circuit denied the government’s attempts to restart the deportations, which a federal judge had blocked earlier in March. On Friday, the Trump administration asked the Supreme Court to allow it to resume the deportations.

At Cecot, as Ms. Noem filmed a video in front of a row of prisoners that were crowded tightly into bunks behind bars, her flashy watch bulged from her wrist, standing out in an austere scene.

The display led to a great deal of criticism on social media from people who questioned the taste of wearing such a pricey watch for the visit. Cecot, which opened in 2023 and was designed to hold as many as 40,000 prisoners, was a signature initiative of Nayib Bukele, the El Salvadoran president who has gained an international reputation for dealing with his country’s gang problem through mass incarceration — a campaign that has been criticized by multiple human rights groups.

In a statement about the watch, Tricia McLaughlin, homeland security’s assistant secretary for public affairs, wrote that Ms. Noem used the proceeds of her books “to purchase an item she could wear and one day pass down to her children.” Ms. McLaughlin did not address the decision to wear that potential heirloom to Cecot.

It is perhaps not a surprise that Ms. Noem, formerly the governor of South Dakota, owns a Rolex — the Swiss brand has been a watch of choice for politicians for decades. Former president Joseph R. Biden Jr., a known watch enthusiast, wore a Rolex Datejust to his inauguration — a choice that led to some criticism from the right. Presidents Trump, Ford and Reagan all wore Rolexes. And even the former Soviet leader Leonid Brezhnev made a concession to the fruits of private industry when he wore a gold Datejust.

According to the watch journalist Brynn Wallner, the founder of Dimepiece, a site for female watch enthusiasts, the Daytona is among the most sought-after Rolexes. First produced in 1963, the watch shot to popularity when Paul Newman started wearing one. Today, the watch is hard to get — buyers typically have to sit on a yearslong wait list to buy it from an official dealer — and as a result, many resort to paying inflated prices on the secondary market.

“If you’re buying it, you’re flaunting the fact that you can even get one,” Ms. Wallner said. “And you probably pay a little more for it than you had to. It’s a flex piece. It’s a signifier of wealth. It’s not subtle at all.”

Paul Altieri, the founder and chief executive of Bob’s Watches, an online marketplace for the resale and trade of watches, agreed.

“Rolex intentionally keeps supplies limited to maintain exclusivity,” he said. “Most customers won’t be offered one unless they have a longstanding relationship with the dealer or are high-priority clients.”

That Ms. Noem’s watch was quickly identified was to be expected. “Watchspotting,” the internet sport of identifying the watches of public figures, has flourished in recent years.

At the Super Bowl in February, enthusiasts immediately identified a Jacob & Co. Caviar Tourbillon on Tom Brady’s wrist, which retails for more than $700,000. Jay-Z was even more extreme at last month’s Grammy Awards, wearing a Patek Philippe Minute Repeater Perpetual Calendarwatch, which retails for more than $2 million. Mr. Trump is often spotted wearing luxury watches beyond just Rolexes, and also has his own line of signature watches that cost as much as $100,000 each.

Watchspotters often pay close attention to any glimpses they can get of watches during awards shows and galas, and they quickly report what they find online.

Now, thanks to Ms. Noem, they have expanded their purview to prisons.