"I think what happened in Mississippi last night definitely sends a strong signal to us in Alabama that until we have party registration that same kind of thing could happen here in our state," Chairman Bill Armistead said, as quoted by AL.com.
Cochran squeaked by state Sen. Chris McDaniel (R-MS) in the runoff partly because his campaign courted both black voters and Democrats. Alabama has similar open primaries, where a voter could technically vote in one party's primary and then cross over to vote in another party's runoff election.
Armistead told AL.com that while he was fine with Republican candidates rallying Democratic support in a general election, he wouldn't have any of it in primaries or runoffs.
"We don't need to have Democrats participating in the Republican primary, and until we have party registration, that's going to happen," he said.
Armistead has an even tougher stance on voter fraud. Earlier this month he announced that the state party would offer a $1,000 reward to anyone with information about suspected fraud that leads to a felony conviction.