The reality of the language is such that this kind of ellipsis is not unheard of, and is indeed rather common. The Corpus of Contemporary American English has 27 cites for "can and have", and the British National Corpus has 8.
Out of these 35 cites in total, at least 10 are actually from academic publications, which is an unusually high proportion, and more still are from newspapers, magazines, or books. Here are a few for your convenience:
Reform plans can and have been constrained (if not defeated) particularly in the foreign policy arena by the preferences of the supreme leader.
Journal of International Affairs, 2003
From a constructionist perspective, however, individuals and entire groups can and have shifted from one category to another, from non-white to white and even on occasion from white to non-white.
Journal of American Ethnic History, 1999
Although federal troops can and have been used to assist local authorities in emergencies when requested by state officials, they are not authorized to enforce local laws.
ABA Journal, 1999
Accidents can and have been defined in terms of the statistics collected by the DTI (see above), and most of these clearly involve an interaction between the individual and the environment.
Current Psychology, 1996
They also can and have been used as rationales for maintaining the status quo of our healthcare system.
Hospital Topics, 1993
The company secretarial department of the firm can and have acted in this role on occasions.
KPMG MAS engagement manual, 1993
There are two broad approaches to inadequacy that can and have been taken.
Public finance and public choice, 1992
Universities can and have contributed to Geological Survey research[.]
Misc unpublished — The purpose and use of the research, undated
Liberation theologians can and have replied, "Don't ask us; we are theologians, not economists or political scientists."
Theological Studies, 1992
They can and have safeguarded the provision of humanitarian aid.
Liverpool Daily Post and Echo, undated
Low-end PC makers can and have shrugged off the three-month delay imposed on the P5.
Unigram x, undated
The character of social classes and class structure, the problems of ideologies and cultural traditions, the emergence of a relatively autonomous state and of the systemic qualities of world economic interactions can and have been informed by Marxist intellectual traditions.
Social Research, 1990
Components can and have been handled where the drawing reference and attitude bear no relationship to the kinematic motion.
The computer-based design process, 1986
Do note the vast variety of fields: history, politics, theology, psychology, medicine, sociology, mechanical engineering, finance.
Now, 35 cites is not much — the corpora have over 1 million cites for can alone, and over 2 million for have —, but it's enough to show that this is not a one-off error, nor a construction limited to careless speech. And to further put the number into perspective, the corpora have only 12 cites for "orange car", and only 7 for "cool beer".
All that said, you are of course still free to omit this kind of ellipsis. And we are free to call it strange, clumsy, or illogical. We just can't call it ungrammatical. Grammatical constructions are under no obligation to be logical, unclumsy, or unstrange.
I won't leave you without a caveat, though: all of the above only holds for can specifically. The corpora barely have a single cite for this construction with any other modal.