Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined as follows:
Its weight filtration satisfies ${\rm W}^{-2}M = 0$ and ${\rm W}^{-1}M = M$, so it is pure of weight $-1$. Its Betti realization is $M_{\rm B} = {\bf R}e_+ \oplus {\bf R}e_-$. If $K=\bf R$ then the archimedean Frobenius $F_\infty$ acts via $F_\infty e_\pm = \pm e_\pm$; write $M_{\rm B}^\pm = M_{\rm B}^{F_\infty=\pm1} = {\bf R}e_\pm$. Its de Rham realization $M_{\rm dR}$is the $K$-subspace of ${\bf C} \otimes_{\bf R} M_{\rm B}$ with basis $\{1 \otimes e_+, i \otimes e_-\}$. Thus if $K=\bf C$ then $M_{\rm dR} = {\bf C} \otimes_{\bf R} M$ and if $K=\bf R$ then $M_{\rm dR} = 1 \otimes M_{\rm B}^+ \oplus i \otimes M_{\rm B}^-$. Its Hodge filtration is ${\rm Fil}^p = M_{\rm dR}$, ${\rm Fil}^{q+1} = 0$, and if $p < k \leq q$ then ${\rm Fil}^k = K \cdot (1 \otimes e_+ + i \otimes e_-)$. Its complex conjugate Hodge filtration $\overline{\rm Fil}^k$ is the same, except changing the last $i$ to $-i$. Of course the Hodge numbers are $p,q$ with multiplicity one each, and its $\gamma$-filtration jumps from $M_{\rm B}$ to $0$ at index $p$.
Such $M(p,q)$ can be made symplectic self-dual (into the Tate object), and conversely every Hodge structure over $K$ with coefficients in $\bf R$, pure and self-dual (into the Tate object), is isomorphic to a sum of copies of $M(p,q)$ for various $p,q$ as above. (Feel free to correct me if I'm wrong about this.)
My question is, what is its central value $\varepsilon$-factor $\varepsilon(M(p,q))$?
I seem to be getting inconsistent answers from the literature. For example, Deligne ("Valeurs de fonctions $L$ et périodes d'intégrales" \S 5.2,5.3) seems to give $(-1)^p$ if $K=\bf R$ and $-i(-1)^p$ if $K=\bf C$, whereas Fontaine and Perrin-Riou ("Antour des Conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions $L$" \S III.1.1.10,III.1.2.7) seems to give $i(-1)^p$ if $K=\bf R$ and $-1$ if $K=\bf C$. My general expectation is that the symplectic self-duality (into the Tate object) should force $\varepsilon(M) \in \{\pm1\}$, which would contradict each author's computation in some way; is this incorrect? Are their formulas genuinely consistent, and I am applying them wrong (perhaps by not specifying something like an additive character or a Haar measure somewhere)? If their formulas are genuinely different, then what is correct?
