Does there exist backwater valves for sink drains?
I have what I think is a standard plumbing set up

It appears that you have the dishwasher drain hose plumbed directly to the disposal instead of to an air gap. ALSO you have a washing machine plumbed to a tail piece with a dishwasher stub,  Both are not &quot;standard&quot;. (In the USA)
Washing machines pump water at a significant rate and may be overwhelming your 1 1/2&quot;  sink plumbing,  2&quot; is code for washing machine drains.
Adding  a check valve is not the correct solution.
The first step it to plumb it properly and then if you still  have the issue   diagnose the problem.