I saw your response soon after you posted it, but had to go to a meeting outside and did not have time. Coming back home, I tried a couple of things and it turns out to be more difficult than I thought.

Although this solution is far from elegant, it seems to work:

  DB<1> $x = q|<blah1 phase="2" type="MyType" more_keys="Values" <Unwa
+nted/>  <SomeTagIwant><k1="v1"></SomeTagIwant>|;

  DB<2> $y = q|<blah1 phase="2" type="MyType" more_keys="Values" <Unwa
+nted/>  <SomeTagIDontWant><k1="v1"></SomeTagIDontWant>|;

  DB<3> print "$_ " for $x  =~ /type="([^"]+)".+?(\w+?TagIwant)  | typ
+e="([^"]+)"/x;
MyType SomeTagIwant
  DB<4> print "$_ " for  $y =~ /type="([^"]+)".+?(\\w+?TagIwant)  | ty
+pe="([^"]+)"/x;;
  MyType
  DB<5>
[download]

It could be slightly improved with a named regex:

  DB<6> $type = qr/type="([^"]+)"/;

  DB<7> print "$_ " for $x  =~ /$type .+?(\w+?TagIwant) | $type /x;
MyType SomeTagIwant
  DB<8> print "$_ " for $y  =~ /$type .+?(\w+?TagIwant) | $type /x;
  MyType
[download]

Well, I think that haukex's solution below looks better.