It may well be, since

print $s . $s . $s;       # still produces the "error",
print '' . $s . $s . $s;  # makes it go away,

# But!

print join '', $s, $s, $s;   # is fine,
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The latter should still be essentially a concatenation in which the first element of the list is the tied variable. So what we're observing should result from an interaction between the tie mechanism and the concatenation operator.

BTW: it is well known that when a bare variable is passed as the first argument of a join, it is re-evaluated every time it's needed, and

print join $s, ('') x 4;
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gives me (with the code of Re^2: tie() weirdness)

FETCH called!
FETCH called!
FETCH called!
FETCH called!
123
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which is unexpected too, but can be understood in terms of an extra-evaluation.