Select elements of a list by comparing it to another list

A rule-based approach:

patt = List /@ OrderlessPatternSequence @@@ Alternatives @@ sub;
Cases[prod, {x_, y : patt} :> {x, y}]

{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}

And a functional approach:

sel = Outer[ContainsAll, Last /@ prod, sub, 1];
Pick[prod, Or @@@ sel]

{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}

A simpler one:

select[{label_, values_}] := MemberQ[sub, {OrderlessPatternSequence @@ values}]
Pick[prod, select /@ prod]

{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}

You can use the same approach with Select:

Select[prod, select]

{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}

You can also turn prod into an Association using sorted second elements as keys and then use Lookup:

Lookup[Sort/@sub] @ GroupBy[ Sort@*Last] @ prod

{{{"x4", {"m", "n"}}}, {{"x5", {"o", "p", "q", "r"}}}, {{"x2", {"e", "f", "g"}}}}

Make it a function:

ClearAll[lookUp]

lookUp[keys_] := Lookup[Sort /@ keys] @* GroupBy[Sort@*Last]

lookUp[sub] @ prod

{{{"x4", {"m", "n"}}}, {{"x5", {"o", "p", "q", "r"}}}, {{"x2", {"e", "f", "g"}}}}

getMatches[prod_, sub_] := Module[{test},
  Scan[(test[Sort[#]] = True) &, sub];
  Cases[prod, {_, y_?test}]]

getMatches[prod, sub]

{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}

Also

getMatches[prod_, sub_] := Module[{test},
  Scan[(test[Sort[#]] = True) &, sub];
  Select[prod, test@*Last]]

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