AnkhSVN and TortoiseSVN use the same administrative data that is stored in your working copy. This store is used for all management, so they will always see each other administrative changes and the design of the subversion core libraries makes sure they can't break the working copy for each other.
Both products have their own status caches to speed up the showing of the glyphs/icons/overlays. These caches are not essential for the functionality of the products but most users prefer them to be accurate anyway. These glyphs can be out of date in both clients, but refreshing will fix them directly.
The latest stable release of AnkhSVN listens to some of the communication between TortoiseSVN and the Windows explorer to keep the status cache even better up to date than the previos version. (In most cases glyphs will now directly change in AnkhSVN.. just like in the Explorer..)
But even before those changes most AnkhSVN users didn't have any real issues using these clients side by side (or even with more clients), as ultimately the source code handling only relies on the administrative data.