Relabelled dice that can hit every sum 2 to 12

A six-sided die has each face labelled with a number from $\{1,2,3,4,5,6\}$; labels may repeat and need not all be used. Two dice with the same face multiset count as the same die. Rolling such a die twice, all faces equally likely, how many different dice make every total from $2$ to $12$ possible?

$11$. Totals $2,3,11,12$ force the faces $1,2,5,6$. Total $5$ (and $9$) then needs a $3$ or a $4$. The sixth face is free, so there are $6$ dice containing a $3$, $6$ containing a $4$, and $1$ (the standard die) containing both: $6+6-1=11$.