The fact that some asset (in this case corporate bonds) has positive correlation with some other asset (equity) doesn't mean buying both isn't a good idea. Unless they are perfectly correlated, the best risk/reward portfolio will include both assets as they will sometimes move in opposite directions and cancel out each other's risk. So yes, you should buy corporate bonds.
Short-term government bonds are essentially the risk-free asset. You will want to include that as well if you are very risk averse, otherwise you may not. Long-term government bonds may be default free but they are not risk free. They will make money if interest rates fall and lose if interest rates rise. Because of that risk, they also pay you a premium, albeit a small one, and should be in your portfolio.
So yes, a passive portfolio (actually, any reasonable portfolio) should strive to reduce risk by diversifying into all assets that it reasonably can. If you believe the capital asset pricing model, the weights on portfolio assets should correspond to market weights (more money in bonds than stocks). Otherwise you will need to choose your weights. Unfortunately we are not able to estimate the true expected returns of risky assets, so no one can really agree on what the true optimal weights should be. That's why there are so many rules of thumb and so much disagreement on the subject. But there is little or no disagreement on the fact that the optimal portfolio does include risky bonds including long-term treasuries.
To answer your follow-up question about an "anchor," if by that you mean a risk-free asset then the answer is not really. Any risk-free asset is paying approximately zero right now. Some assets with very little risk will earn a very little bit more than short term treasuries, but overall there's nowhere to hide--the time value of money is extremely low at short horizons. You want expected returns, you must take risk.