Sharing products from a catalogue point of view is possible across stores in Magento. Note that stock/inventory will also be shared across both sites.
The challenge you'll have is that carts can't be shared across domains because of the way sessions/cookies are managed. If this was a priority you would instead need to create an extension that posted cart contents from one site to the other and the cart was recreated so that carts were synchronised across sites. This is easier for logged in customers but guest customers would have to go through your store switcher code in order for this to work. If they visited the other site manually, the cart synchronisation process wouldn't function.
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