**Core Concept**
The question requires knowledge of breast cancer types and their characteristics. Breast cancer can be classified into various subtypes based on histological and molecular characteristics. Bilateral breast cancer refers to the occurrence of cancer in both breasts.

**Why the Correct Answer is Right**
Invasive lobular carcinoma (ILC) is a subtype of breast cancer that originates in the lobules of the breast and can spread to surrounding tissues. It is known to have a higher propensity for bilateral involvement compared to other subtypes. This is due to the fact that ILC often arises in the terminal duct-lobular units (TDLUs) of the breast, which are present in both breasts. The molecular characteristics of ILC also contribute to its bilateral nature.

**Why Each Wrong Option is Incorrect**
* **Option A:** Invasive ductal carcinoma (IDC) is the most common subtype of breast cancer, but it is not typically bilateral. IDC arises in the ducts of the breast and is often unifocal.
* **Option B:** Ductal carcinoma in situ (DCIS) is a non-invasive form of breast cancer that is confined to the ducts. While DCIS can be bilateral, it is not typically considered a carcinoma in the same sense as invasive cancers.
* **Option C:** Phyllodes tumors are rare breast tumors that can be benign, borderline, or malignant. They are not typically bilateral and are not classified as carcinomas.

**Clinical Pearl / High-Yield Fact**
It is essential to note that bilateral breast cancer can be a manifestation of genetic syndromes such as BRCA1 and BRCA2 mutations. Patients with a personal or family history of breast cancer should be screened for these genetic mutations.

**Correct Answer:** C.