Duncan Crabtree-Ireland, the executive director and chief negotiator for the actors’ union, has spent the past two decades toiling behind the scenes during contract talks. The spotlight, he knows, is for the SAG-AFTRA president, usually a well-known performer like the current office holder, Fran Drescher.

But ever since the guild went on strike on July 14 for the first time in 40 years, things have been different.

In the past three months, Mr. Crabtree-Ireland, 51, has stepped out from behind the negotiating table and made fiery speeches, walked film festival red carpets and reached out to the union’s younger members via Instagram reels. His more frequent appearances have given people ample opportunity to see the tattoos on his forearms, a visual clue to how much the professional and the personal are intertwined for him. On the right are five symbols — a record, a play button, a film reel, a megaphone and a radio antenna — representing the contracts he’s negotiated for…