You can use firring strips if you just want to cross-brace between joists.
To brace the structure agaist swaying , put diagonals at 45 degrees between the beams and the posts at the corner posts. the further down the post or the more corners you brace, the steadier it will be. For appearance, you may want to use braces the same size as the posts.
Diagonal cables from the bottom of one post to the top of another would also work, but you would need then to "criss-cross". This can cut off access, but would help if you want to train some vegitation in one or more bays in each direction.
Apparently, you do not have the posts into the ground far enough to provide rigidity.