You don't say what kind of K-1 you have. A 1065 K-1 is not the same as an 1120-S K-1. But I looked on both of those forms and line 13 is for "other credits" on both forms. It has nothing to do with passive income that would get reported on page 2 of the SCH E on your personal 1040 tax return.
whereas I am told (as active income) it should show up on Schedule E.
No, that's wrong. Only passive income gets reported on SCH E. Line 13 on both forms is not for passive income.
Specifically, the amount shown on Line 13H of the K-1 is reported on Schedule A Line 9, rather than Schedule E, page 2, Line 28 (Passive Income and Loss).
It sounds like investment interest expense which should appear on both Schedule A (Line 9) and Schedule E, page 2, Line 28. Note that the expense is only deductible to the extent of investment income.