How to calculate temporary/periodic similarities of an increasing series in real time?
well, This is an interesting question! Let's formulate it again to be precise in answer:


  I have two real-time streams of values. There are temporary similarities (transient similarities) between two. How to capture it?


As you are talking about similarity, I did not limit the scope to correlation. One may define another measure of similarity. I would say you should define your similarity function first (you may choose correlation. I do as well for my example) and a period of inspection (it's simply a window of length $m$ ending at current timestamp).

Using these two and starting from $m+1$th sample, you can calculate your similarity measure (here correlation) as a time-series. This time-series starts after $m$th sample of input series so keep it in mind to keep similarity measure and original timestamps synchronized. The interval between two consecutive points in output (similarity series) is defined based on the resolution you need. after $m+1$th sample you can shift your window on step at time and calculate the similarity accordingly. Then the intervals in output will be as same as input. 

Of course it keeps you more away from real-time as the frequency and number of calculations is maximum. You can define a margine for yourself (let's say $d$ sample) and calculate similarity every $d$ time. Then you lose some info (discarding the samples in between) but you get closer to real-time (you will have $d*n$ seconds to perform computation). 

Please let me know if it need clarification. Welcome to the community BTW :)