Magically reappearing fringes in Michelson interferometer for large path length differences? I've built a simple Michelson interferometer from two mirrors and a beam splitter according to the following schematic:

Image source
My setup differs from the one in the publication in that mirror M1 can be both translated along the optical axis as well as tilted. This is in order to adjust the optical path length difference between both interferometer arms, i.e.

*

*from the beam splitter to mirror M1 and back,

*from the beam splitter to mirror M2 and back.

Merely tilting M1 would allow for only a very limited path length difference, so I used a sliding mount to increase the adjustment range.
Yesterday, I tested the setup using the expanded and collimated beam from a regular green laser pointer, starting off from nearly identical lengths of the interferometer arms. The tilt of mirror M1 was adjusted to get a nice fringe pattern with the right distance between fringes to be comfortably observed by eye on a nearby screen. The fringes have very nice contrast going to zero in the minima.
Next, I wanted to determine the laser pointer's coherence lengths. So I started to translate mirror M1 and thereby increasing the optical path length difference. As one would expect, the fringe contrast started to decrease at some point, until they disappeared completely.
Now comes the strange part: When I continued to translate the mirror even further, at some point, the fringes reappeared and I got the nice, high-contrast pattern I started with. Confused, I translated the mirror even further, and the fringes disappeared again...but after some distance, again, they reappeared. This continued to happen within the range of motion of mirror M1, which is about 3-4 cm (= 6-8 cm range of adjustable optical path length difference, counting the roundtrip to and from the mirror).
What is going on here??
As far as I understand, one is not supposed to see fringes reappearing as the laser pointer has only a limited coherence length. Reason: When the optical path length difference between the interferometer arms exceeds that coherence length, all stable phase correlation is lost (which is exactly how one can measure the coherence length and therefore coherence time/spectral width of a light source using a Michelson interferometer).
So how is it possible that the phase correlation is "magically" recovered for increasing path length differences?
UPDATE 1: The proposed explanations consider the laser pointer to be multi-mode, which is probably true. But it's not clear (at least to me), how the presence of multiple wavelengths can explain the reappearance of interference fringes way beyond the expected coherence length of the source. Typical laser diodes have bandwidths of tenths of nanometers, corresponding to several hundreds of micrometers in coherence lengths. The optical path length difference adjusted here, however, are in the order of centimeters.
But what's more important: A superposition of waves with different frequencies can never produce stationary interference patterns. Such superposition results in beating, i.e. amplitude modulations, which are not stationary, but propagate through space.
Through the discussions in the comment section, I've come to the conclusion that this question requires a more rigorous, mathematical explanation. Either that or a small simulation script to visualize the observed effect. Since this requires some more thought/work, I'm starting a bounty.
UPDATE 2: Since the question of coherence length couldn't be answered here, a follow-up question has been posted, where this problem is discussed more rigorously.
 A: Your laser is producing light of at least two wavelength, say $\lambda_1$ and $\lambda_2$, which are fairly close together.
The condition for the first extinction and then reappearance is $2d=m\lambda_2=(m+1)\lambda_1\Rightarrow \Delta \lambda = \lambda_2-\lambda_1 = \lambda_1/m$ where $m$ is the order when the fringes are fully seen again.
So you start with the interferometer set at the zero order (found by using white light) and then you change a path length whilst counting the number of fringes which have passed until the fringes reappear again.
This counting process is obviously difficult when you cannot see the fringes so you can do it by estimation knowing how many fringes pass for a known value of mirror movement $d$.
On the other had you can use the relationship $\Delta \lambda \approx \frac{{\lambda_1\lambda_2}}{2d}$.
This is a standard procedure in an undergraduate lab to find the separation of the two Sodium D lines.

There may well be more than two wavelengths which would make it an even more interesting practical exercise.
