I stumbled upon the following proposition, and haven't found an error in my proof yet.
By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition.
Let $X$ be a locally connected space. If $U$ and $V$ are open, open-unicoherent subsets of $X$ such that $X = U \cup V$ and $U \cap V$ is connected, then $X$ is open-unicoherent.
This is similar to the van Kampen theorem but its statement is purely point-set. I wonder if this (or its negation) is a known result, and in case it isn't, if this is something someone has interest in. I'm aware of the result of Pierce (the paper 'Special Unions of Unicoherent Continua') that if $X$ is a Hausdorff continuum, $A$ and $B$ are closed unicoherent subsets of $X$ such that $X = A \cup B$ and $A \cap B$ is connected and locally connected, then $X$ is unicoherent. Actually, I asked this question about 5 years ago. Unless I'm mistaken, the proof of the above proposition is pretty simple and fits in a few pages.
Edit: In the paper 'Incidence Relations in Unicoherent Spaces', Stone proves that for locally connected spaces, unicoherence is equivalent to the "open-unicoherence". So, what I (think I) proved is:
Let $X$ be a locally connected space. If there are open unicoherent subsets $U, V \subset X$ covering $X$ such that $U \cap V$ is connected, then $X$ is unicoherent.
This is similar to the result of Pierce, but differs in some of the conditions. I wonder if I can also assume that only the intersection is locally connected.
