Neil, what you got there is a feature request, which around here they call 'Ideas'. You post them over here. I say this because Tableau definitely does not currently work this way. Now if you decide to give up your zoom controls and use action filters you can get your two charts to sync up.
What would be the best way to achieve this using action filter?
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Neil, actions are fairly straight-forward, but you do have a lot of options, so best to read the manual. Here's a link to the Actions section. You'll want to focus on the ones in yellow:
Once you understand them, they can be very useful.