Out of curiousity, are you still only taping the pep? Tape won't add any strength to the joins, they'll still be able to bend. Glue might be the answer.
Is there any flat edges on the bottom of the pieces? I'm referring to joining them both together, back & front, then try & sit them flat. That way when they try & collapse downwards, you'll have the card's strength to support them. If you do both halves seperately, then lay flat on the "join" they don't have any vertical support, if you know what I mean.
Of course I may be misunderstanding something also.