I am using ArcMap 10.3.1 and there are 4 feature classes in question involved in this mapping project (above and beyond the basemap itself).
The data frame's projection is NAD_1983_2011_StatePlane_California_V_FIPS_0405_Ft_US.
The first layer is hydrography data courtesy of the USGS. The geographic coordinate system is GCS_North_American_1983 and the projection is not defined. Since this feature class was part of an extant geodatabase, I exported the data into the geodatabase that I created for the current mapping project, redefined the GCS, and assigned a projection. The GCS is currently GCS_NAD_1983_2011 and the projection is NAD_1983_2011_StatePlane_California_V_FIPS_0405_Ft_US. As far as I can tell, assigning a projection didn't shift anything since the features from the original feature class and the projected feature class coincide spatially.
The second feature class consists of polygons representing the designated wilderness areas across the southern half of the U.S., so the spatial extent is greater than any of the rest of the layers with the exception of the basemap. This layer has a GCS, GCS_North_American_1983, but no projection, so I used the Define Projection tool to assign a projection. Before I did that however, I exported the features that overlap with the national forest in question and created a new layer out of those features. (Before this point, using the Define Projection tool yielded wacky results with features that were nowhere near accurate, so I had to dump the layer at least 2x to compensate for my mistakes.) The GCS of the new layer is currently GCS_NAD_1983_2011 and the projection is NAD_1983_2011_StatePlane_California_V_FIPS_0405_Ft_US.
The next layer contains parcel data: Projected Coordinate System: NAD_1927_StatePlane_California_V_FIPS_0405 Geographic Coordinate System: GCS_North_American_1927 I used the Project tool to change both the GCS and the projection to match the data frame, so the GCS of the reprojected layer is currently GCS_NAD_1983_2011 and the projection is NAD_1983_2011_StatePlane_California_V_FIPS_0405_Ft_US.
Last but not least, the last layer contains boundary data for the national forest I am focusing upon. Projected Coordinate System: NAD_1927_California_V Geographic Coordinate System: GCS_North_American_1927 Again, I used the Project tool to change both the GCS and the projection. Again, the new layer's GCS is currently GCS_NAD_1983_2011 and the projection is NAD_1983_2011_StatePlane_California_V_FIPS_0405_Ft_US.
The end result is that the boundaries of the wilderness area do not coincide with the parcel data. The gap between the two layers is such that the parcel data is shifted approximately 200 feet to the north and 200 feet to the west. I've dumped the data from these last two layers more times than I bothered to track, each time trying something slightly different in the settings of the tools in the hope of yielding a better result.
I did apply transformations and just to make sure that they would be applied, I double checked the transformations in the data frame, deleted the 4 layers using ArcCatalog, and then re-imported the parcel boundaries, wilderness boundaries, and the national forest boundary one by one. Since the 4th layer, the water features, were already a part of a separate geodatabase, I dragged and dropped the layer and then then exported the data into the geodatabase I created for the project. I mostly did this just so that I could visually verify whether there was a positional shift after the projection. (There wasn't.)
The two layers using GCS_North_American_1927 - the parcel boundaries and the national forest - had this transformation applied: NAD_1927_To_WGS_1984_79_CONUS + WGS_1984_(ITRF08)_To_NAD_1983_2011. The remaining two layers - the water features and the wilderness boundaries - were both originally in GCS_North_American_1983 and neither feature class had a projection applied until the data was either imported or exported into my project's geodatabase. The water features and the wilderness boundaries had this transformation applied: WGS_1984_(ITRF00)_To_NAD_1983 + WGS_1984_(ITRF00)_To_NAD_1983_2011.
Again, the boundaries of the wilderness areas still do not coincide with the parcel data.