You need to specify the field for each site individually:
orderby="site1_shortname:event_date|site2_shortname:event_date" sort="asc|asc"
What's happening right now is that it's matching the field name event_date to site 1 and ordering by that field for site 1, but using the default orderby (entry_date) for site 2.
(Docs.)
However, thinking about this further, this is actually expected behaviour. When specifying multiple orderby fields, it's sequential. So first you're ordering by field_id_X (whatever event_date is in site1). Well, site 2's entries are 0 in that field, so they come first.
You'd have to use a common field, like entry_date, to order entries from both sites sequentially together. (Check out EEvent Helper, which allows you to clone a custom date field to the Entry Date, among other things.)