  We consider the signature rank of the units in real multiquadratic fields.
When the three quadratic subfields of a real biquadratic field $K$ either (a)
all have signature rank 2 (that is, fundamental units of norm $-1$), or (b) all
have signature rank 1 (that is, have totally positive fundamental units), we
provide explicit examples to show there exist infinitely many $K$ having each
of the possible unit signature ranks (namely signature rank 3 or 4 in case (a)
and signature rank 1,2, or 3 in case (b)). We make some additional remarks for
higher rank real multiquadratic fields, in particular proving the rank of the
totally positive units modulo squares in such extensions (hence also in the
totally real subfield of cyclotomic fields) can be arbitrarily large.
