The correct way to handle it is to have a common product backlog as both teams are working in the same product. Also both teams should have the same definition of done or at least close enough to make sure it's compatible. In order to adapt your DoD to this new scenario, you will need to change to "done, working and integrated".
There will always be dependencies between teams no mater how you manage it. So a good way to manage it is by making one of the teams responsible for the dependencies management. This doesn't mean that the other team do not care, it just means that one of them will be actively looking for issues and solutions.
For sprint planning, a good practice may be to have a first common sprint planning where the PO share the vision of the sprint and defines goals for all the teams. Once done, each team may go for a separated sprint planning where they do the normal story pointing and estimation of work to be done in the print.
On the daily standup, the integration team may change a bit the 3 questions to answer to include information about what happened with integration yesterday and the day to come.
On the sprint review, all teams may be together showing the work done, working and integrated. On the other hand, the sprint retrospective may be done within each separate team.
Hope it helps