The question reads:
I gather from the answer to the linked question that what happens in principle is that the police would investigate, and the local prosecutor would file charges against Alice.
This is not correct even in principle. At least in the united-states, the local prosecutor might or might not file charges.
Reasons why a case might not be prosecuted
There are many reasons why a prosecutor might not file charges, even is s/he believed that the accused did the crime, quite aside from any outright bribery.
Perhaps the most important and most common is that the prosecutor thinks that a trial will not result in a conviction. No prosecutor wants to spend time, effort, and resources only to have a jury acquit. Also, win percentages are an important way that prosecutors are evaluated, either by their bosses (for assistants) or the voters.
Similarly, if the crime is fairly minor, the prosecutor may be unwilling to spend limited resources on the case, especial;y if a jury trial seems likely.
If the prosecutor feels that a conviction would work an injustice, s/he may choose not to proceed.
If a prosecutor thinks that pursuing the case will be politically unpopular, s/he may well choose not to proceed. Local DAs at the state level are elected officials in most states, and while Federal prosecutors (US Attorneys and their assistants) are appointed, they are often very alive to political forces -- the post of US Attorney is often a stepping stone to elected office. (For example former NJ Governor Christie was a US Attorney before being elected Governor.)
Options for a victim or interested party
One can, of course, request the local prosecutor to proceed with a case. This can be backed with a campaign to draw public attention, and hence political influence, to the case. The prosecutor can always decided to proceed with a case (unless a statute of limitations has expired).
In many US states (for example MA) the state Attorney General has the option to take over any particular case from the local DA, and have his or her office handle it instead. One could ask the AG to exercise this power.
In a Federal case, the US Attorney General could direct a US Attorney's office to proceed with a case, although that is quite unusual. Again public/political pressure mi8tght well matter.
If the prosecutor had actually accepted a bribe, that would be a crime. If this could be proved, the prosecutor should be removed from office and quite probably convicted. In such a case the new prosecutor would be very likely to review the case with a very different eye.
As other answers have discussed, Bob might be able to file a civil suit.
If the actions Alice is accused of constitute both a state and a federal crime, Bob could ask the other jurisdiction to take the case.
When jury trials were first introduced in England, prosecutions were normally started by a private person, through an action known as Appeal (This is different than the modern use of "appeal" to mean bringing a case to a higher court.) But the King and the central authorities under the king found that often crimes were not prosecuted which they wanted addressed, particularly when the accused was a wealthy or powerful person, or the potential accuser was weak or poor.
So the Grand jury was created. This was originally not a protection against the state, but an instrument of the state (the crown) to help determine who committed crimes and to get prosecution started against them, even when no one would file an Appeal.
Public prosecution and private prosecution by Appeal coexisted in England until 1819 when the Appeal of Murder etc Act, 1819 was passed, which eliminated private appeal of crime. This was a direct result of the case of Ashford v Thornton, the previous year. In that case the accused, Thornton (who had previously been acquitted on a public prosecution) elected to defend himself by "wager of battle" -- that is a trial by combat -- and was held to be entitled to do so.
See The Law of the Land by Charles Rembar for coverage of the above history, written for the non-lawyer.
David Friedman, in chapter fifteen of Legal Systems Very Different From Ours discusses in detail 18th Century England's then-current system of private prosecution, largely from an economic point of view, pointing out its advantages and disadvantages, and some of the side effects.
In thew US, private prosecution has never been much favored, and even where it is legally possible, it has remained rare.