Text typed into different apps (e.g. Chrome, Sublime) find functions (Cmd-F) overwrites each other

Question

I'm seeing unusual behavior in the interaction between some Mac programs like Firefox, Safari, Sublime, and Chrome. I'll concentrate on the last two. I have a Macbook Pro (Retina, 13-in, mid-2014) running macOS 10.13.4; Chrome 66.0.3359.139 (Official Build) (64-bit); and Sublime 3.0 (build 3143).

Suppose I try searching for some text in a document using Sublime by pressing Cmd-F and typing code. The text code appears in Sublime's Find box (bottom of window) and the window scrolls to highlight over the first instance in the document. Then I switch focus to Chrome using Cmd-tab and try searching for some text on a webpage by pressing Cmd-F and typing webtext. The text webtext appears in Chrome's Find box (upper-right corner) and the window scrolls to highlight over the first instance in the webpage. Seem's OK so far.

However, when I switch focus back to Sublime, I find that webtext is now written in Sublime's Find box, erasing my previous search term code. If I type in a new search like code2 into Sublime's Find box, switching back to Chrome reveals it has likewise overwritten webtext there.

If I try repeating this with both the Chrome window and Sublime window visible at the same time, I find that the Find-box overwriting in the non-focused program doesn't occur until I switch focus to it with Cmd-Tab.

What's going on, and how do I prevent this behavior?

(None of this depends on whether I use keyboard shortcuts like Cmd-Tab or Cmd-F vs. selecting things from the menu with the mouse. I see the same universal interaction with Safari and Firefox, but not Mathematica and Evernote. Presumably there's some sort of OS-wide tool the offenders are using, and Mathematica and Evernote have custom searches.)