Can Vim remember modes when switching between tabs?

I originally noticed this in MacVim, but turns out this happens in the
terminal as well. If you have two tabs open and enter the Insert mode
in one, then switch to the other tab (by clicking on it in GUI, for
example), the Insert mode is still active. This wreaks havoc when I'm
editing something and want to quickly switch to the other tab to
search/look up/etc, because the Normal mode keystrokes get inserted
into the buffer.

Any workarounds for this?

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