So what have the seemingly amicable exes been up to lately during their family vacation with daughter Apple and son Moses?
"They were spotted in town yesterday doing some shopping," a local source exclusively tells E! News. "They bought some bracelets for their kids. They were trying to keep a low profile. Martin was seen holding his daughter's hand and Gwyneth was seen holding the son's hand."
However, our eyewitness did point out that "their demeanor was not that of a happy couple," adding, "They seemed very nonchalant, with no signs of affection being shown between them. But they are very affectionate with their children."
Earlier this week, local sources confirmed exclusively to E! News that Paltrow and Martin had rented an upscale home on Eleuthra Island, a property that typically leases for $5,000 to $6,000 per week.
We were informed that they were spotted having dinner on Tuesday evening, with a local resident telling us, "The island is a place where you go not to be seen. People come to the island to really get away. It is a very private island and there is nothing commercial here. It's a lot of land without a lot of people living here."
Meanwhile, Paltrow took a moment early Friday morning to break her silence regarding the separation via her lifestyle website Goop. After writing a short paragraph in relation to chicken recipes, she told readers, "P.S. CM and I in deep gratitude for the support of so many."