Thanks Adam, that worked for me too. Yes, I'd clicked the Reproject option rather than the Repair option.
FYI, all our SDE layers are set to the ESRI out-of-the-box British National Grid coord system (description shown below.
WKID: 27700 Authority: EPSG
Linear Unit: Meter (1.0)
Geographic Coordinate System: GCS_OSGB_1936
Angular Unit: Degree (0.0174532925199433)
Prime Meridian: Greenwich (0.0)
Semimajor Axis: 6377563.396
Semiminor Axis: 6356256.909237285
Inverse Flattening: 299.3249646
If I add an SDE layer to Manifold it recognises the coord system to be ARCSDE:2. If I then overlay it on the basemap, which was originally built using the same layer, then they don't align. If I take a copy and run the repair coord system then it does match, the same as the shapefile.
Like I say, if I overlay the same SDE layer (or shapefile) in QGIS (or Arc) then they align without requiring any manual correction. Maybe they adjust the layer automatically?