SYDNEY, Australia — Australia’s governing Liberal Party met for hours behind closed doors on Tuesday to debate same-sex marriage, twisting itself in knots and ending up with a two-step approach: If the Senate does not open the polls for a compulsory national vote, party officials said, then Australians will be invited to cast voluntary ballots on the issue by mail as early as September.

Prime Minister Malcolm Turnbull, speaking at a news conference after the discussions, said that holding a postal vote if the Senate rejected an in-person one would give people the opportunity to “have their say,” though neither vote would be legally binding.

After months of delays and rumors about how Australia would make a democratic decision about the contentious issue, the process itself has become a subject of widespread scorn.

While other countries have decided on same-sex marriage in the courts or in Parliament, lawmakers in Australia have spent months bickering about mechanisms. The disagreements include whether to put the issue to legislators or voters, and whether that should be through the mail or a national vote, known in Australia as a plebiscite.