What would an anydice function to calculate odds on Daggerheart's duality dice look like?

If you're just interested in the total, you can use a function to replace the result with a high value if the two dice are equal, and return the simple sum otherwise, like this:

function: daggerheart hope H:n fear F:n bonus B:n {
 if H = F {
  result: 99
 }
 result: H + F + B
}

output [daggerheart hope d12 fear d12 bonus 5 + d6]

Here the number parameter type instructs AnyDice to run the function for all numbers that the dice could possibly produce.

If you want to know whether the roll was with Fear, Hope, or Critical, you can use a function like this:

function: daggerheart hope H:n fear F:n {
 if H = F {
  result: 2
 }
 if H > F {
  result: 1
 }
 result: 0
}

output [daggerheart hope d12 fear d12]

where Fear is 0, Hope is 1, and Critical is 2.

You can even sort of combine the two by putting the total in the 1s and 10s place and the Fear/Hope/Critical in the 100s place, like this (this time the example uses a d20 for Hope):

function: daggerheart hope H:n fear F:n bonus B:n {
 if H = F {
  result: 299
 }
 if H > F {
  result: 100 + H + F + B
 }
 result: H + F + B
}

output [daggerheart hope d20 fear d12 bonus 5 + d6]

Turn the table sideways and you have a rough idea of the joint distribution.

The left "hump" (i.e. values of 0-99) is the totals with Fear, the right "hump" (i.e. values of 100-199) is the totals with Hope (plus 100) and 299 is the Critical.