  The product of two real spectral triples {A1,H1,D1,J1,gamma1} and
{A2,H2,D2,J2(,gamma2)}, the first of which is necessarily even, was defined by
A.Connes as {A,H,D,J(,gamma)} given by A=A1 x A2,H=H1 x H2, D=D1 x I2 + gamma1
x D2, J=J1 x J2 and by, in the even-even case, gamma=gamma1 x gamma2.
Generically it is assumed that the real structure J obeys the relations
J^2=epsilon Id, JD=epsilon' DJ, Jgamma = epsilon'' gammaJ, where the
epsilon-sign table depends on the dimension n, modulo 8, of the spectral
triple. If both spectral triples obey Connes' epsilon-sign table, it is seen
that their product, defined in the straightforward way above, does not
necessarily obey this epsilon-sign table. In this note, we propose an
alternative definition of the product real structure such that the epsilon-sign
table is also satisfied by the product.
