Just 41 percent say there should be a religious exemption to that requirement. That’s a shift since an earlier AP-GfK poll conducted in July. Then, Americans were about evenly split, with 49 percent saying officials with religious objections should be exempt from issuing marriage licenses to same-sex couples and 47 percent saying they should be required to issue them.

The July poll was conducted just after the Supreme Court ruled that all states must allow gay and lesbian couples to marry legally. In the intervening months, Kim Davis, a county clerk in Rowan County, Ky., became a household name after being jailed for contempt of court over her refusal to issue marriage licenses to same-sex couples.

That shift was especially stark among Republicans. A majority of them —58 percent — still favor religious exemptions for officials issuing marriage licenses, but that’s down 14 points since 72 percent said so in July. That shift comes despite no equivalent change in Republicans’ opinions on same-sex marriage more generally.