First step: Do nothing. You've got two working, independent products, and that's okay. You can keep running like this indefinitely so there's no need to rush.

Second step: Figure out what the future combined product/company is going to look like. There's no sense doing any integration work until you know where you're going.

Third step: Form a combined team and come up with a plan to build the new product, reusing and rewriting the existing technology as appropriate. The details here entirely depend on the existing products and the end goal.

Keep in mind this situation can easily become political and toxic -- in fact, I would say that's the default outcome -- and will require strong leadership to navigate well.