Ended up solving this a different way. I stashed the IDs for each relationship and then did a regex match with switchee:
{!-- Store the ids for the playa company relationships --}
{exp:stash:set name="board_memberships_ids" parse_tags="yes"}
,{employee_board_memberships:child_ids delimiter=","},
{/exp:stash:set}
{exp:stash:set name="investments_ids" parse_tags="yes"}
,{employee_investments:child_ids delimiter=","},
{/exp:stash:set}
{!-- Set the related companies list --}
{exp:playa:children field="employee_board_memberships|employee_investments"}
{exp:stash:append_list name="employees_companies" parse_tags="yes"}
{stash:item_class}
{switchee variable="stash:board_memberships_ids" parse="inward"}
{case value="#,{entry_id},#"}board-members{/case}
{/switchee}
{switchee variable="stash:investments_ids" parse="inward"}
{case value="#,{entry_id},#"}investments{/case}
{/switchee}
{if company_investment_stage == "public"}ipos{/if}
{if company_investment_stage == "acquired-merged"}acquisitions{/if}
{/stash:item_class}
{stash:item_sort}{title}{/stash:item_sort}
{stash:item_content}
<a href="/portfolio/{url_title}" class="modal-trigger">{title}</a>
{/stash:item_content}
{/exp:stash:append_list}
{/exp:playa:children}