I've settled on something I like for the coat of arms of the country
of Silvia, as opposed to the archdiocese. It's significantly younger than the diocesan arms, for one because the archdiocese existed before the County of Tarentaise was combined with it, and for two because the archbishops didn't at first make a strong distinction between their spiritual and temporal powers.
I blazon it as follows:
Quarterly: First and fourth sable, a bend chequy gules and argent of two tiers; second gules, a cross argent; third or; over all an escutcheon gules, two keys in saltire or interlacing a wheel of twelve spokes argent.
To unpack the symbolism:
The arms is divided into four quarters, the top left and bottom right quarters showing a red and white checkered diagonal strip over a black field. This is the Cistercian bend, a reference to two figures of Silvish history who belonged to the Cistercian religious order: St. Bernard, after whom the Little St. Bernard Pass is named; and St. Silvius, after whom the country is named.
The top right quarter, a white cross on a red field, is the coat of arms of the House of Savoy. Relations between Silvia and Savoy were always delicate at best because the latter regularly challenged the archbishops' power. But there is no denying the deep interconnection of the two states. For example, the archbishops often interceded as arbitrators in Savoy's other territorial disputes. And more than a few Savoyard sons have served as archbishop.
The bottom left quarter, a gold field, is the arms of Pope Alexander III, who expanded the authority of the archbishop-counts to the borders that would become modern Silvia.
The archdiocesan coat of arms is placed over top of all of these.