[SalesForce] Is it possible to Upsert a List with external ID specified

This should work:

List lst_PartnersAcc = [SELECT CUST_NUM__c
                                             FROM SYSTEMS__c WHERE Account__c IN (Select  AccountId from AccountShare where ( UserOrGroupId IN : map_group OR  UserOrGroupId =:userId )AND RowCause IN ('Territory', 'TerritoryManual', 'TerritoryRule'))
                                             ];
Set tempList = new Set();
for(Systems__c s : lst_PartnersAcc) {
    tempList.add(s.Cust_Num__c);
}
List orders = [select id,name from orders__c WHERE customer_1__c IN :tempList OR customer_2__c IN :tempList ETC...
                                             ];

Just needed to add the Customer Number to the set individually for each Systems__c.

I know this approach is strange because you are still working with a List<SObject>, but when you assign it you can make it more specific (e.g. List<Account>) by using Type.forName and Type.newInstance methods.

public static void dynamicUpsert(List<SObject> records)
{
    Schema.SObjectType sObjectType = records.getSObjectType();
    if (sObjectType != null)
    {
        String listType = 'List<' + sObjectType + '>';
        List<SObject> castRecords = (List<SObject>)Type.forName(listType).newInstance();
        castRecords.addAll(records);
        upsert castRecords;
    }
}

The getSObjectType call may not be completely reliable, especially since some of these records are being inserted and hence won't have ids. For that reason, it is probably better to accept sObjectType as an additional parameter instead of trying to determine it on the fly.

public static void dynamicUpsert(List<SObject> records, SObjectType sObjectType)
{
    String listType = 'List<' + sObjectType + '>';
    List<SObject> castRecords = (List<SObject>)Type.forName(listType).newInstance();
    castRecords.addAll(records);
    upsert castRecords;
}

Update

Some bad news, as discovered here (actually earlier than your post), the above methodology does not always work. Specifically, if I want to perform a partial upsert and also specify an external Id field, I get a compile fail.

Database.upsert(castRecords, externalIdField); // compiles, throws TypeException
Database.upsert(castRecords, /*allOrNone*/ false); // compiles, throws TypeException
Database.upsert(castRecords, externalIdField, /*allOrNone*/ false); // compile fail 

Additional Update

I have a case open with support and it has been escalated to Tier 3. Currently the responses I'm getting are all along the lines of "will update you tomorrow," but I will post here if I get any resolution.

Additional Update

Support claims the three parameter signature failure is WAD. They said if I want them to actually fix it, I need to post on an Idea, so here it is, vote for it!

Tier 3 ... performed some testing and it looks like that we cannot do upsert with List regardless of which of the 3 method signatures you use.

The only difference in behavior seems to be that we block using the 3-arg one at compile time and the 2-arg and 0-arg ones fail at run- time.

Basically for the 3-arg we validate at compile time that you are not passing in a list parameterized with SObject. But for the other 2 we compile it and we allow it to run and check in the call whether the list is generic.

We could make the 3-arg work but it seems like a feature request and based on the behavior of the 0-arg and 2-arg versions we would presumably still enforce that the underlying list isn't generic.