Scrum is a lightweight, open, easy-to-understand, proven-to-work, team-based, result-oriented product development process.
There will not be enough arguments against having you and your offshore team start with Scrum.
It will bring in needed transparency, healthy rhythm, start building trust and domain understanding. It will help you structure status meetings and necessary conference calls.
It is probably the easiest but robust process you can start with. Read a mini-book "Scrum and XP from the trenches" by Henrik Kniberg to get ideas how it feels.