if the adjective of the noun BACTERIUM is 'bacterial' and that of VIRUS is 'viral', what is the adjective for (BACTERIO)PHAGE? Is 'phagial' accepted?
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It is bacteriophagous as given in wiktionary.
From bacterio- (“bacteria”) + -phagous (“eating”).
In the literature, it's most common to see hyphenated compound adjectives, like "phage-mediated," "phage-dependent," or "phage-encoded" where "bacterial" or "viral" might be analogous. For example, "Phage‐mediated selection and the evolution and maintenance of restriction‐modification."
Sometimes "phage" seems to be used as a standalone adjective, like in "Phage antibodies: filamentous phage displaying antibody variable domains." Although I'm not sure that it would be correct to use "bacterial antibodies" or "viral antibodies" in this case, so maybe that one is an anomaly.
I would suggest following one of these trends, unless you want to do a lot of explaining to reviewers.