Go to your ICA, or agreements that are part of the job assignment The key is not to be identified as a mystery shopper. If either client says no, you cannot do it, then ethically you should not do it. If it is not forbidden, then are you at risk as being identified as a MS?
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Do not read so much, look about you and think of what you see there.
Richard Feynman-- letter to Ashok Arora, 4 January 1967, published in Perfectly Reasonable Deviations from the Beaten Track (2005) p. 230