Stereoisomerism in organic chemistry arises due to the presence of stereocenters or double bonds with restricted rotation. Stereoisomers have the same molecular formula and connectivity, but differ in the arrangement of their atoms in space.

Given compound: 1-chloro-2-(3-chlorocyclobutane) ethene

Step 1: Identify stereocenters or double bonds with restricted rotation
The given compound has a double bond between carbon 1 and carbon 2. The carbon atoms in the cyclobutane ring are also stereocenters, as they have four different substituents attached to them.

Step 2: Determine the number of stereoisomers
The number of stereoisomers can be determined using the following formula:
Number of stereoisomers = 2^n, where n is the number of stereocenters or double bonds with restricted rotation.

In the given compound, there are two stereocenters in the cyclobutane ring and a double bond between carbon 1 and carbon 2. Therefore, the number of stereoisomers = 2^2 x 2^1 = 4 x 2 = 8.

However, some of these stereoisomers may be identical due to symmetry. In this case, the cyclobutane ring has a plane of symmetry, which means that some of the stereoisomers are identical to their mirror images. Therefore, the actual number of different stereoisomers is less than 8.

Step 3: Identify identical stereoisomers
The two stereocenters in the cyclobutane ring can have two different configurations: cis or trans. However, the cis and trans stereoisomers are mirror images of each other and are therefore identical. This reduces the number of stereoisomers to 4.

Step 4: Final answer
Therefore, the number of different stereoisomers for 1-chloro-2-(3-chlorocyclobutane) ethene is 4. The correct answer is option B.