My Sony Xperia Active has the same issue since it came back from repair with Android 2.3.4 and firmware 4.0.2.A.0.62 installed. Needs to have BT turned on all the time for e.g. MyTracks to connect, while the ANT+ Demo App works all the time. This is strange, since MyTracks' code is based on ANTPlusDemo's AntPlusManager.java.
[UPDATE]: The problem here is opening channels before receiving the ANT_ENABLED_ACTION message. This was OK before ANT Radio Service v3.4, but with the update Sony phones 'may take time to re-enable on app-start', as the release note states.