Howard,
I understand your concern about not having enough material on a 4x4 to but joint the two beams. I'm guessing 2x8's????
You can do a couple of things here. Attach 2x4's to the tops of both sides of your 4x4 posts to make it three inches wider. Either lag them on or carriage bolt them together. This will give you allot more surface area to attach the 2x6 beams. To make this look attractive, cut the bottoms at 45 degrees so it looks like the 4x4 post gets wider at the top. You can do this with two on each side if you like, or even get creative and make some sort of pattern.
The other option is to buy some metal T brackets, or large plating. Attache it to the post first, then to your 2x6 beams.
Eddie