Couch and her teenager diving partner, Lois Toulson, of City of Leeds Diving Club, were in with a chance of a podium finish with one round to go.
Diving last, they needed a score of 82 points to following in the footsteps of Tom Daley and Dan Goodfellow and claim a medal. But their final effort was their most disappointing of the night, with Toulson struggling with her execution, and they only scored 63.36 for their three-and-a-half inward somersault.
It meant they scored 319.44 points and finish fifth behind winners China, Malaysia, Canada and North Korea.
It was the same placing Couch had achieved at the last Olympics with Sarah Barrow.
But the 27-year-old, competing at her third Olympic Games, was really positive about the outcome.
“We’re over the moon,” said Couch about their performance. “That was actually a personal best score for us.
“We have only been diving together for eight months and I can honestly say that is the most enjoyable competition I have been in. I really enjoyed it.”
The big talking point of the night was the colour of the diving pool in Rio. The previous two days it had been the normal blue colour but for the women’s 10m synchro final it had turned a dark green.
Couch said: “When we went up to 10m we could not believe how green it was, but it didn’t matter. I found it was easier to spot. We did pretty well and got a personal best, so maybe we should ask for green water from now on.”
Plymouth Diving’s Couch will now turn her attention to the individual competition, which starts on August 17. She will be joined in that event for his club colleague Barrow.