I'm so sorry to say this, but we still need more facts. On its face, a provision stating, "Tenant will not obstruct or interfere with the rights of other tenants or occupants of the Property" does not equal "[you] can't operate a business that is in competition with another business in the same shopping center," unless you (the bakery) knew that the sandwich shop, in fact, had a right to be the sole business, operating in this shopping center, that sold sandwiches, in which case, the landlord MIGHT be able to use this provision against you. You really need to consult with a local attorney (and both of the other 2 attorneys who answered your question, from earlier today, are really excellent).
Disclaimer: The materials provided below are informational and should not be relied upon as legal advice.
It would be hard to answer your question without reviewing your lease and related paperwork in their entirety. The answer would also depend on whether your business or the competitor first became a tenant at that property, what the landlord knew about your intended business before you moved in, what limitations the lease places (other than the language you quote above) on the types of businesses you may run, and whether any friction exists between you and the landlord. If you are prepared to retain an attorney, feel free to contact our office. Be sure to consult your own attorney to protect your legal rights.
I do not think this provision would limit your ability to sell sandwiches. The answer to your problem depends on a review of the entire lease. I would start by reviewing the "Permitted Use" section that states what purposes you can use the leased space for. If it says "bakery" and nothing more and nothing in the lease excludes selling sandwiches, you are in good shape. You need to consult with an attorney who has experience in leasing.
Possibly, the sandwich shop tenant has an exclusive in their lease (giving them the exclusive right to sell sandwiches in the center). In that case, they would have a claim against the landlord, not you. In such a case, the landlord should have negotiated a provision in your lease excluding the sale of sandwiches.
Although a further review of the lease terms would be needed, the provision you quoted does not mean that you will not compete with any other business. Typically the provision you quoted is used to mean that your business will not interfere with others by making too much noise, or emitting foul smells, or blocking access, for example, that negatively impact other tenants in the shopping center. I would recommend that you show these Avvo responses to your Landlord, and if that does not get him or her to back off this matter, you should retain counsel to assist you.
This response does not create an attorney-client relationship and is not intended to provide legal advice for your specific situation.