cmd-tab behavior on Mavericks with multiple displays
I believe this coincides with the Dock's location.  Just tested it on my MacBook Air, connected to a non-Apple external display, and whenever I moved the Dock from screen to screen the Application Switcher would follow.

You can summon the Dock on your big display by dragging the cursor to the bottom of it's display, essentially dragging down at the bottom.  After a second the Dock should pop up.  Once the Dock is on the desired display press commandtab to summon the Application Switcher.