Are Frequently Bought Together's discounts compatible with other app discounts or automatic discounts?

Yes, as long as all the discounts involved have been configured as combinable and can be combined with each other. You can check under what circumstances discounts can be combined at this link:  https://help.shopify.com/en/manual/discounts/discount-combinations

To configure Frequently Bought Together's discounts as combinable, please refer to the related article.

The following image shows a Frequently Bought Together discount combined with a Shopify automatic discount:

Please note that discounts are not combinable by default, so every time a discount is applied to an order, it erases any previous discount.

For that reason, if you have any other app that creates discounts or you have configured automatic discounts in your store, they might interfere with each other.

If you add a discounted bundle with Frequently Bought Together and then another app adds another discount, the Frequently Bought Together discount will be removed.

Likewise, if another app adds one discount to the shopping cart and then a discounted bundle is added by Frequently Bought Together, the other app discount will be removed.

Same happens if a discount code is entered manually during the checkout process, it erases all previous discounts:

Finally, automatic discounts override discount codes so, if the order matches the criteria of a configured automatic discount, the Frequently Bought Together discount will be removed.