Can I have detailed balance without reverse updates? I am implementing a Markov Chain Monte Carlo algorithm and want to obtain a stationary distribution.  In order to do this I want to have detailed balance, as this is a sufficient, although not necessary condition for obtaining a stationary distribution. If there is a solution to the following problem, but it is not called detailed balance anymore I am fine with that as well.
I have many different states, and types of updates, and so far I have made sure that each update has a reverse, such that I can use the detailed balance criterion for determining acceptance probability for the updates. Now I want to get a bit crazy (lazy), and drop the reverse update, such that I can move
$A \rightarrow B$, but not $B \rightarrow A$. Instead I will make it possible to go $B \rightarrow C \rightarrow A$.
I think this should be possible, I just have to choose my acceptance criterion carefully. The weight of the states are simple, so now I need to consider the update suggestion probabilities. Is it sufficient to choose $P(A\rightarrow C)\cdot P(C\rightarrow A)$ or maybe I need to consider intermediate steps and this is to simple and naive. Is there any way to solve this or will I have to make a reverse update?
Background: My motivation for asking is that the update $A \rightarrow B$ is very simple, I am just deleting a site $i$ with a continuous parameter $t$. $B \rightarrow A$ would include making a site with the specific parameter $t$. This is not impossible, but it would feel nice to be able to just make a default site $B \rightarrow C$ and then choose the parameter $t$ afterwards $C \rightarrow A$. Also it would be cool to increase my understanding of my approach a little bit.
 A: If I understand your question the answer is yes, that is possible. 
You have the correct intuition. The detailed balance is actually an overly strict condition. For a Markov chain, it is enough to use the global balance condition:
$$\sum \limits_x \pi(x) P(x\to x') = \sum \limits_{x'} \pi (x') P(x' \to x)$$ 
For some stationary distribution $\pi(x)$ and some transition probability $P(x \to x')$. 
Compare that to the detailed balance condition, which requires that every pair of states $x,x'$ have no net flow between them. 
$$\pi(x) P(x \to x') = \pi (x') P(x' \to x)$$
Similar to what you proposed, under global balance you know that total transitions in and out of some state $x$ sum to zero, but there may be some net flow between any pair of states $x$ and $x'$. For example, you could have an update that moves $A \to B$ with no reverse update $B \to A$ so long as it is possible to return to $A$ by some other path like $A \to B \to C$. 
There are MC methods that use global balance. And these methods can be very efficient. But here's the catch: it's really hard to guarantee that you are enforcing global balance, and if you mess up, it might cause an error that is very hard to detect. As a result, most methods use detailed balance. 
Global balance methods:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.120603
https://arxiv.org/abs/1309.7748
