Can’t figure out query logic

If I follow your description right you want posts which have any $typeIDs, but all of $unionIDs ?

Aside from wrong meta_query choice, you miss operator argument which specifies which kind of match you want.

I think your query should be something like this:

'tax_query' => array(
    'relation' => 'AND', // you want both conditions to match
    array(
    'taxonomy' => 'audition_type',
    'terms' => $typeIDs,
    'operator' => 'IN', // this is default, just showing you difference
    ),
    array(
    'taxonomy' => 'union_requirement',
    'terms' => $unionIDs,
    'operator' => 'AND', // not default, we want matches to all terms
    )
),

See Taxonomy Parameters in Codex for further documentation.