Cursor disappears when other windows are open

I'm working on a game where an important part of the UI is dependent on customizing the cursor. Every single set of instructions I've seen for making a custom cursor in Unity includes setting Screen.showCursor to false. The problem is that if I have the game running on a web browser in one tab and open another tab at the same time, the cursor still disappears over the square of screen that was running the game in the first tab. Any suggestions for fixing this?