Can someone introduce a vector that simultaneously introduces two or more antibiotic resistances (markers) for better screening of resistant bacteria that lack the vector?
I am using pET Duet-1 for protein expression, and this vector has two selectable markers. I think you may go with this vector. I have got good results, but if you are doing subcloning, then I suggest you go with pUC19.
It certainly is possible, the cloning vectors that were used in the very early days (like pBR322) had 2 antibiotic resistant markers although usually you screened for clones having lost one of them. But most people don't find it necessary any more to do that as the single resistance markers work very well. If you are having trouble with clean selections then the problem may be elsewhere.
What selection are you using? Some antibiotics are notorious for "satellite" colony formation, like ampicillin. You can use a more stable form like carbenicillin. Make sure your media is made fresh (less than 4 weeks old) & you should be just fine with one marker. Using 2 is very old-school and not needed.