PROBLEM 1:
A linked file of existing conditions has been created with all objects on "New Construction". Everything in the linked file appears as New Construction.
CAUSE:
Revit forces you to find a match for each and every phase. If there isn't a matching phase Revit will automatically put your linked model into both Existing AND New Construction.
SOLUTION:
The only workaround that I've found so far is this.
Go into the model that will be linked into your current project.
Create a new phase at the "future" end of the phasing, label Future Blank (or something else).
Map phases:
LINKED FILE > HOST FILE
Existing > Existing
New Construction > Existing
Future Blank > New Construction
PROBLEM 2:
Objects in linked files are not displaying correctly.
This is even though Visibility / graphics settings say that the host's Phase Filter is being used.
CAUSE:
Linked files use their own Phase Filter settings. They ignore the host file's settings.
SOLUTION:
Open the linked file and change Phase Filter settings to match the settings in the host file.