What’s a Couple to Do When One Works for Donald Trump and the Other for Elon Musk?

What does a married couple do when their bosses — the president and the world’s richest man — break up?

Such is the conundrum facing Stephen and Katie Miller, a pair of ride-or-die Trump loyalists who have seemingly found themselves on opposite sides of the Donald-Elon breakup.

Mr. Miller is a top aide to the president, a deputy chief of staff for policy and an immigration hard-liner, while Ms. Miller had until recently been serving as an adviser and a spokeswoman for the Department of Government Efficiency, or DOGE, Elon Musk’s cost-cutting initiative.

Last week, as Mr. Musk was leaving Washington behind, having been handed a golden key by the president and displaying a mysterious black eye, the Millers also found themselves in the spotlight. There were news reports that Ms. Miller was also leaving DOGE to work for Mr. Musk full time, a development that led to speculation about her relationship with the administration — and her husband.

That chatter only intensified on Thursday when Mr. Musk and Mr. Trump engaged in an hourslong, insult-dappled spat, complete with Mr. Musk suggesting the president was ungrateful for the $250 million he spent on getting him elected and Mr. Trump saying he could cancel the mogul’s many government contracts. (As with any squabble, there were other brickbats and accusations, too.)

By Thursday afternoon, that kerfuffle had seemingly spilled over into the Millers’ universe again, when it was reported that Mr. Musk had stopped following Mr. Miller, and several other conservative leaders, on X, the platform he owns.