The relation of a person you share a half-sibling with
Solution 1:
Stepbrother/Stepsister
Paraphrasing Dictionary.com:
The son (/daughter) of one's stepparent who is not the son (/daughter) of either of one's biological parents. The stepson (/daughter) of one's parent who is not one's half-brother (/sister).
This is used when parents with existing children remarry, even if no "bridge" half-sibling comes along later; and there's no point in making up another term.